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Z.^       ^-Tyt- 


V.  ^^^/V     /^/ 


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University  of  California  •  Berkeley 


THE 


i^i 


GYROSCOPE. 


BY  MAJOR  J.  G.  BARNARD,  A.  M., 


CORPS    OF    ENGINEERS,     U.     S.     A. 


m 


FROM    Barnard's    American    journal   of    education. 


NEW   YORK:    D.    VAN    NOSTRAND. 
HARTFORD:    F.    C.    BROWNELL. 

1858. 


IfflfflP^^PPBfflPMffiMl 


Mil 


mfi 


^^^iPiiPi'^^ 


ERRATA. 

In  the  first  paper  of  this  pamphlet  the  references  to  pages  should,  wherever 
met  with,  read  instead  of  "  52, "  "  53, "  "  54,  " — "  540,  "  "  541,  "  "  542.  " 

In  the  second  paper,  for  "spiral"  '^' and  "spiral  motion,  "  read  "helix" 
and  helical  motion. " 


f^O.   f^^     ^-.^^^^*t^3^     <Q^€JU 


THE   PHENOMENA 


THE  GYROSCOPE, 


WITH   TWO   SUPPLEMENTS, 


THE  EFFECTS  OF  INITIAL  GYRATORY  VELOCITIES,  AND  OF  RETARDING 
FORCES  ON  THE  MOTION  OF  THE 

GYROSCOPE. 


BY  MAJOR  J.   G.  BARNARD,  A.  M., 

COKPS    OF    ENGINEERS,     U.     S.    A. 


PROM    Barnard's    American   journal    of    EnucATioN. 


NEW   YORK:    D.    VAN    NOSTRAND. 
HARTFORD:    F.   C.    BROWNELL. 

1858. 


PREFATORY  REMARKS. 


No  physical  phenomenon  has  ever  more  highly  excited  the  curiosity  of  the 
public  generally  than  that  exhibited  by  the  simple  instrument  known  as  the 
♦'  Gyroscope."  None,  based  so  directly  upon  the  very  fundamental  laws  of  me- 
chanics, (viz.,  those  which  refer  to  inertia^  and  to  gravitation,)  seems,  at  first 
sight,  to  exhibit  so  utter  a  violation  of  them. 

It  is  not  indeed  the  unskilled  in  mechanics  alone  who,  seeing  an  apparent 
suspension,  in  this  little  instrument,  of  the  very  first  law  governing  matter  which 
addresses  itself  to  the  experience  of  childhood,  is  perplexed. 

The  scientific  man  too,  the  mathematician  (unless  his  studies  have  happened 
to  lead  before  in  this  very  direction,)  is  startled,  and  is  prone  to  ask  himself  if 
so  paradoxical  a  phenomenon  does  not  involve  some  new  and  hitherto  unknown 
mechanical  principle,  or  some  modification  of  those  already  admitted. 

Yet  there  can  be  perhaps  no  more  beautiful  illustration  of  those  laws,  no 
more  convincing  proof  of  their  absolute  truth,  and  adequacy  to  explain  all 
purely  mechanical  phenomena,  than  is  found  in  the  solution  of  the  problem  of 
the  Gyroscope. 

To  exhibit  this  perfect  harmony  of  the  phenomenon  with  laws  universally 
known  and  understood  (so  far  as  the  primal  laws  of  matter  can  be  understood) 
has  been  the  governing  idea  in  my  mind  in  preparing  these  pages ;  and  auxili- 
ary to  this,  I  desired  to  set  at  rest  a  vexed  question,  and,  while  correcting  the 
numerous  errors  which  had  been  circulated  in  popular  and  even  scientific 
journals,  to  place  the  analysis  of  the  problem  in  such  a  form  that  all  who  had 
so  much  knowledge  of  mechanics  as  may  be  derived  from  text-books,  could 
follow  it. 

If  I  had  addressed  mathematicians  alone,  and  sought  results  merely,  it  is 
proper  to  say  that  I  could  have  arrived  at  them  by  much  shorter  methods. 

To  those  who  seek  a  popular  explanation  and  do  not  find  satisfactory  that 
which  I  strive  to  give,  independently  of  the  analysis,  in  the  latter  part  of  my 
first  paper,  I  can  only  say  that  all  attempts  at  a  purely  popular  explanation  I 
have  yet  seen  have  been  failures,  and  that  the  perplexity  of  terms,  rather  than 
the  intrinsic  difficulty  of  the  subject,  renders  such  explanations  of  little  avail  to 
those  who  cannot  also  comprehend  the  analysis. 

The  two  supplementary  papers  of  this  pamphlet  became  necessary  in  order 
to  apply  the  theory  to  the  actual  circumstances  under  which  the  Gyroscope  is 
seen  ;  the  more  so  because  at  the  first  glance  the  actual  motions  of  the  instru- 
ment seem  as  paradoxically  to  violate  the  theory,  as  the  theoretical  motions 
seem  to  do  the  laws  of  nature. 

The  theory  of  the  Gyroscope  contained  in  these  pages  is  not  new,  nor  does 
it  profess  to  be  so:  but  the  whole  constitutes,  so  far  as  I  know,  the  only  thorough 
application  of  the  laws  of  rotary  motion  to  all  the  observed  phenomena  of  the 
instrument,  involving,  as  they  do,  the  effects  of  friction,  resistance  of  the  air 
and  of  initial  gyratory  velocities. 

J.  G.  B. 

New  York,  April  21st,  1858. 


CONTENTS. 

1.  The  Self-sustaining  Power  of  the  Gyroscope  Analytically  Examined. 
From  Barnard's  American  Journal  of  Education  [No.  9]  for  June,  1S57,  pp» 

537-550.      ^:,^(:l) 

2.  On  the  Motion  of  the  Gyroscope  as  modified  by  the  retarding  forces 

OF  friction,  and  the  resistance  of  the  air  :  WITH  A  brief  Analysis 
OF  the  Top. 
FVom  Barnard's  American  Journal  of  Education  [No.  11] /or  December,  1857, 
pp.  529-536. 

3.  On  the  Effects  of   Initial    Gyratory    Velocities,  and  on  Retarding 

Forces  on  the  Motion  of  the  Gyroscope. 
From  Barnard's  American  Journal  of  Education  [No.  13]  for  June,  1858,  pp 
299-304. 


^.u^y.    /SS^,  jr^y 


XX.    EDUCATIONAL  MISCELLANY  AND  INTELLIGENCE. 

KOTARY  MOTION  AS  APPLIED  TO  THE    GYROSCOPE. 

BT  HAJO&  J.  Q.  BABNAKD,  A.  M. 

Corps  of  Engineers  of  United  States  Army. 


After  reading  most  of  the  popular  explanations  of  the  above 

Ehenomenon  given  in  our  scientific  and  other  publications,  I 
ave  found  none  altogether  satisfactory.  While,  with  more  or 
less  success,  they  expose  the  more  obvious  features  of  the  phe- 
nomenon and  find  in  the  force  of  gravity  an  efl&cient  cause  of 
horizontal  motion,  they  usually  end  in  destroying  the  founda- 
tion on  which  their  theory  is  built,  and  leave  an  effect  to  exist 
without  a  cause  ;  a  horizontal  motion  of  the  revolving  disk  about 
the  point  of  support  is  supposed  to  be  accounted  for,  while  the 
descending  motion,  which  is  the  first  and  direct  effect  of  gravity 
(and  without  which  no  horizontal  motion  can  take  place),  is 
ignored  or  supposed  to  be  entirely  eliminated.  Indeed  it  is 
gravely  stated  as  a  distinguishing  peculiarity  of  rotary  motion, 
that,  while  gravity  acting  upon  a  non-rotating  body  causes  Tt 


to  descend  vertically,  the  same  force  acting  upon  a  rotary  body" 
causes  it  to  move  horizontally,  A  tendency  to  descend  is  supposed 
to  produce  the  effect  of  an  actual  descent ;  as  if,  in  mechanics, 
a  mere  tendency  to  motion  ever  produced  any  effect  whatever 
without  that  motion  actually  taking  place. 

Whatever  *  mystification'  there  may  be  in  analysis — however 
it  may  hide  its  results  under  symbols  unintelligible  save  to  the 
initiated,  it  is  most  certain  that  the  greater  portion  of  the  physi- 
cal phenomena  of  the  universe  are  utterly  beyond  the  grasp  of 
the  human  mind  without  its  aid.  The  mind  can — indeed  it 
must — search  out  the  inducing  causes,  bring  them  together  and 
adjust  them  to  each  other,  each  in  its  proper  relation  to  the  rest ; 
but  farther  than  that  (at  least  in  complicated  phenomena)  un- 
aided, it  cannot  go.  It  cd^nnoi  follow  these  causes  in  all  their  va- 
rious actions  and  re-actions  and  at  a  given  instant  of  time  bring 
forth  the  results. 

This,  analysis  alone  can  do.  After  it  has  accomplished  this, 
it  indeed  usually  furnishes  a  clue  by  which  to  trace  how  the 
workings  of  known  mechanical  laws  have  conspired  to  produce 
these  results.  This  clue  I  now  propose  to  find  in  the  analysis 
of  rotary  motion  as  applied  to  the  gyroscope. 


538  J.  G.  BARNARD    ON   THE    GYROSCOPE. 

The  analysis  I  shall  present,  so  far  as  determining  the  equations 
of  motion  is  concerned,  is  mainly  derived  from  the  works  of 
Poisson  (vide  "Journal  de  I'Ecole  Polytech."  vol.  xvi — Traite  de 
Mecanique,  vol.  ii,  p.  162).  Following  his  steps  and  arriving 
at  his  analytical  results,  I  propose  to  develop  fully  their  mean- 
ing, and  to  show  that  they  are  expressions  not  merely  of  a  visi- 
ble phenomenon,  but  that  they  contain  within  themselves  the 
sole  clue  to  its  explanation :  while  they  dispel  all  that  is  myste- 
rious or  paradoxical,  and,  in  reducing  it  to  merely  a  "particular 
case"  of  the  laws  of  "rotary  motion,"  throw  much  light  upon 
the  significance  and  working  of  those  laws. 

Although  not  unfamiliar  to  mathematicians,  it  may  not  be 
uninteresting  to  those  who  have  not  time  to  go  through  the  long 
preliminary  study  necessary  to  enable  them  to  take  up  with 
Poisson  this  special  investigation ;  or  whose  studies"  in  mechan- 
ics have  led  them  no  farther  than  to  the  general  equations  of 
"  rotary  motion"  found  in  text  books,  to  show  how  the  particu- 
lar equations  of  the  gyroscopic  motion  may  be  deduced. 

In  so  doing  I  shall  closely  follow  him ;  making  however  some 
few  modifications  for  the  sake  of  brevity  and  of  avoiding  the 
use  of  numerous  auxiliary  quantities  not  necessary  to  the  limited 
scope  of  this  investigation. 

The  general  equations  of  rotary  motion  are  (see  Prof.  Bart- 
lett's  "  Analytical  Mechanics"  Equations  (228),  p.  170) : 


(1.) 


In  the  above  expressions  the  rotating  body  (of  any  shape) 
A  B  CD  (fig.  1)  is  supposed  retained  by  the  fixed  point  (within 
or  without  its  mass)  0.  Ox^  Oy  and  Oz  are  the  three  co-ordi- 
nate axes,  fixed  in  space,  to  which  the  motion  of  the  body  is  re- 
ferred. OcCj,  Oy,,  O2,,  are  the  three  principal  axes  belonging 
to  the  point  0,  and  which,  of  course,  partake  of  the  body's 
motion.  The  position  of  the  body  at  any  instant  of  time  is 
determined  by  those  of  the  moving  axes. 

A  J  -Sand  C  express  the  several  "moments  of  inertia"  of  the 
mass  with  reference,  respectively,  to  the  three  principal  axes 
Ox^  Oy^  Oz^\  iV,,  i/j  and  L^  are  the  moments  of  the  accelerat- 
ing forces,  and  Vx^  Vy,  v^,  the  components  of  rotary  velocity,  all 
taken  with  reference  to  these  same  axes. 

Like  lineal  velocities,  velocities  of  rotation  may  be  decomposed 
— that  is,  a  rotation  about  any  single  axis  may  be  considered  as 


J.  G.  BARNARD  ON  THE  GYROSCOPE. 


539 


tlie  resultant  of  components  about  other  axes  (which  may  always 
be  reduced  to  three  rectangular  ones) :  and  by  this  means,  about 
whatever  axis  the  body,  at  the  instant  we  consider,  may  be 
revolving,  its  actual  velocity  and  axis  are  determined  by  a 
knowledge  of  its  components  v^j  Vy,  v^,  about  the  principal 
axes  Ox ,  Oi/^  Oz^^  these  components  being,  as  with  lineal  ve- 
locities, equal  to  the  resultant  velocity  multiplied  by  the  cosine 
of  the  angles  their  several  rectangular  axes  make  with  the  re- 
sultant axis. 

As  the  true  axis  and  rotary  velocity  may  continually  vary,  so 
the  components  Vj;,  Vy,  v^j  in  equations  (1)  are  variable  functions 
of  the  time. 


Fig.  1. 


-V/^ 


^,-^,^ 


xJ^ 

a::^:^^^;^^^  -'Mx 

<iil2v 

\    -"'"'    — -^ 

^/" 

V*^^^-^^^**-*'. /«y7^v/.  ^^ 

/y 

-\y 

^^^K 

" 

For  the  purpose  of  determining  the  axes  Ox^^  Oy^  and  O2,, 
with  reference  to  the  (fixed  in  space)  axes  Ox,  Oy^  Oz,  three  aux- 
iliary angles  are  used. 

If  we  suppose  the  moving  plane  of  cc,  ^/n  at  the  instant  con- 
sidered, to  intersect  the  fixed  plane  of  cc  ?/  in  the  line  N'N'  and 
call  the  angle  xON=ip,  and  the  angle  between  the  planes  xy 
and  x^y^  (or  the  angle  sOz,)=<9,  and  the  angle  NOx^=(p^  (in 
the  figure,  these  three  angles  are  supposed  acute  at  the  instant 
taken,)  these  three  angles  will  determine  the  positions  of  the  axes 
Oic^,  Oz/j,  Oz^j  (and  hence  of  the  body)  at  any  instant,  and  will 
themselves  be  functions  of  the  time ;  and  the  rotary  velocities 
^x,  Vy,  V;,,  may  be  expressed  in  terms  of  them  and  of  their  dif- 
ferential co-efficients. 

For  this  purpose,  and  for  use  hereafter  in  our  analysis,  it  is 
necessary  to  know  the  values,  in  terms  of  gi,  0  and  v,  of  the  co- 


540  J.  G.  BARNARD  ON  THE  GYROSCOPE. 

sines  of  the  angles  made  by  tlie  axes  Ox^,  Oy^  and  Oz^  with 
the  fixed  axes  Oz  and  Oy. 

These  values  are  shown  to  be  (vide  Bartlett's  Mech.,  p.  172)  ^ 
cos  x^  02=  —  sin  6  sin  q>  cos  x^  Oyrrrcos  6  cos  ip  sin  9-- sin  V  cos  (p 

cos  yj  Oz=z-— sin  6  cos  q>  cos  y^  Oy=:cos  ^  cos  V  cos  9)-f-sin  y^  sin  9 

cos  2;  J  0^=     cos  ^  cos  0j  Oy^rsin  0  cos  v 

The  differential  angnlar  motions,  in  the  time  dt,  about  the 
axes  Ox  J,  Oy^j  Oz^^  will  be  Vj;dtj  Vyd%  and  V;2C?^.  We  may  de- 
termine the  values  of  these  motions  by  applying  the  laws  of 
composition  of  rotary  motion  to  the  rotations  indicated  by  the 
increments  of  the  angles  <9,  9  and  V- 

If  6  and  9  remain  constant  the  increment  d^j  would  indicate 
that  amount  of  angular  motion  about  the  axis  Oz  perpendicular 
to  the  plane  in  which  this  angle  is  measured.  In  the  same  man- 
ner dcp  would  indicate  angular  motion  about  the  axis  Oz ,  ;  while 
dd  indicates  rotation  about  the  line  of  nodes  ON.  In  using 
these  three  angles  therefore,  we  actually  refer  the  rotation  to  the 
three  axes  Oz^  Oz^,  ON,  of  which  one,  Oz,  is  fixed  in  space, 
another,  Oz , ,  is  fixed  in  and  moves  with  the  body,  and  the  third, 
ON,  is  shifting  in  respect  to  both. 

The  angular  motion  produced  around  the  axes  Ox^,  Oy^,  Oz^, 
by  these  simultaneous  increments  of  the  angles  9,  ^  and  v^,  will 
be  equal  to  the  sum  of  the  products  of  these  increments  by  the 
cosines  of  the  angles  of  these  axes,  respectively,  with  the  lines 
Oz,  Oz,  and  ON 

The  axis  of  Oz^  for  example  makes  the  angles  ^,  0°  and  90^ 
with  these  lines,  hence  the  angular  motion  v^  dt  is  equal  (taking 
the  sum  without  regard  to  sign)  to  cos  6 dip-]- dtp. 

In  the  same  manner  (adding  without  regard  to  signs), 
Vxdt=QO^  Xy  Oz(iv^+cos  (pdd 
and  Vy  dt= cos  ?/  j  Oz  c?  v^ + cos  (90° + 9)  <^  G- 

But  if  we  consider  the  motion  about  Oz^  indicated  by  dqy,  posi- 
tive, it  is  plain  from  the  directions  in  which  9  and  v  are  laid  off 
on  the  figure,  that  the  motion  cos  OdH^  will  be  in  the  reverse  di- 
rection and  negative,  and  since  cos  6  is  positive  c?V^  must  be  re- 
garded as  negative,  hence 

Vzdt=d(p— cos  Odip. 

The  first  term  of  the  value  of  Va;dt,  cos  x^Ozd^p  [since  cos  a:;,  Oz 
(=— sin  6  sin  9)  is  negative  and  c?v  is  to  be  taken  with  the 
negative  sign]  is  positive.  But  a  study  of  the  figure  will  show 
that  the  rotation  referred  to  the  axis  Ox , ,  indicated  by  the  first 
term  of  this  value,  is  the  reverse  of  that  measured  by  a  positive 
increment  of  6  in  the  second,  and  hence,  (as  cos  9  is  positive,)  dd 
must  be  considered  negative.  Making  this  change  and  substi- 
tuting the  values  given  of  cos  x^  Oz,  cos  y,  Oz,  and  for  cos  (90° 
-hqp),— sin  (f,  we  have  the  three  equations 


J.  G.   BARNARD    ON   THE    GYROSCOPE.  641 

V;rC?^=:siii  6  sin  q>dip—cos  <pdd\ 
Vydt=sm  0  cos  <pd  ip-\'Sm  q)dOy    (2.)* 
Vzdt=d(p— cos  6 dip  ) 

The  general  equations  (1.)  are  susceptible  of  integration  only 
in  a  few  particular  cases.  Among  these  cases  is  that  we  con- 
sider, viz.,  that  of  a  solid  of  revolution  retained  by  a  fixed  point 
m  its  axis  of  figure. 

Let  the  solid  A  B  CD  (fig.  1)  be  supposed  such  a  solid,  of 
which  Oz ,  is  the  axis  of  figure.  It  will  be,  of  course,  a  princi- 
pal axis,  and  any  two  rectangular  axes  in  the  plane,  through  0 
perpendicular  to  it,  will  likewise  be  principal.  By  way  of  de- 
termining them,  let  Ox^  be  supposed  to  pierce  the  surface  in 
some  arbitrarily  assumed  E  point  in  this  plane.  Let  G  be  the 
center  of  gravity  (gravity  being  the  sole  accelerating  force). 
The  moments  of  inertia  A  and  B  become  equal,  and  equations 
(1.)  reduce  to 

Cdv,  =  0  ) 

Advy^(C-A)v^V:,dt—yaMgdt      V    (8.)t 
Advx-\-{C—A)vyVzdtz=z^yhMgdt ) 

in  which  the  distance  OG  of  the  point  of  support  from  the  cen- 
ter of  gravity  is  represented  by  y,  g  is  the  force  of  gravity,  M 
the  mass,  and  a  and  b  stand  for  the  cosines  a;,  O2  and  y^  Oz  and 
of  which  the  values  are  (p.  52) 

a=z  —  sin  6  sin  9,  bz=z  —  sin  6  cos  cp. 

The  first  equation  (3)  gives  by  integration  Vz  =n,  n  being  an 
arbitrary  constant ;  it  indicates  that  the  rotation  about  the  axis 
of  figure  remains  always  constant. 

Multiplying  the  two  last  equations  (3)  by  Vy  and  v^  respect- 
ively and  adding  the  products,  we  get 

A  {Vy  d  Vy  -\-Va:  d  Va:)=:Y  Mff  {aVy—hvx)  dt. 

From  the  values  of  a  and  b  above,  and  from  those  Vx  and  Vy 
(equations  2)  it  is  easy  to  find 

(avy  — bvx)dt=i —sin  0d6z=d. cos  6'^ 
substituting  this  value  and  integrating  and  calling  h  the  arbitrary 
constant 

A  {vy  ^-{-Vx  2)=:2  y  Mg  cos  6-^h  (a) 

*  To  avoid  the  introduction  of  numerous  quantities  foreign  to  our  particular  in 
vcstigation  and  a  tedious  analysis,  I  have  departed  from  Poisson  and  substituted  the 
above  simple  method  of  getting  equations  (2.),  which  is  an  instructive  illustration 
of  the  principles  of  the  composition  of  rotary  motions. 

f  See  Bartlett's  Mech.  Equations  (225)  and  (118)  for  the  values  of  ij  Jtfj  iVj  : 
in  the  case  we  consider  the  extraneous  force  P  (of  eq.  118)  is  ^r;  the  co-ordinates 
x' ,y'  of  its  point  of  application  G  (referred  to  the  axes  Oxy,  Oy^,  Oz^,)  are  zero 
and  z^=zOG=i/:  cosines  of  a,  p  and  7  are  a,  b  and  c:  hence  £^=0,  Mi=']/aMffy 
2^1^ — ybMg. 


542  J.   G.  BARNARD  ON   THE    GYROSCOPE. 

Multiplying  the  two  last  equations  (3),  respectively,  by  h  and  a 
and  adding  and  reducing  by  the  value  just  found  of  c?.cos  6  and 
of  Vz,  we  get 

A{hdvy+adVj:)-\-{C-A)nd.Qo^ez:zO  (6) 

Differentiating  the  values  of  a  and  h  and  referring  to  equations 
(2)  it  may  readily  be  verified  (putting  for  v^  its  value  n)  that 

dhz=z{yx  cos  6  —  an)dt 

daz=z{bn  —  Vy  cos  d)dt 

and  multiplying  the  first  by  Avy  and  the  second  by  Ava;^  and 

adding 

A(vydb-\-Va;da)z=:An{bvj,;  -^avy)dt=z  — And. cos  6. 

Adding  this  to  equation  (Z>),  we  get 

Ad.{hVy-\-aVx)-\- On d . cos  0^=: 0,  the  integral  of  which  is 

A  (bvy-^aVj;)-\-Cn  cos  6z=:l  [I  being  an  arbitrary  constant),     (c) 

Eeferring  to  equations  (2)  it  will  be  found  by  performing  the 

operations  indicated,  that : 

2   .         2         •    2/J^'^V ^^^ 

bvy'{-ava:=z  —  sm^d-— 

Substituting  these  values  in  equations  (a)  and  (c),  we  get 

Cn  cos  6- A  sin^  d ~=l 
dt 

If,  at  the  origin  of  motion,  the  axis  of  figure  is  simply  de- 
viated from  a  vertical  position  by  an  arbitrary  angle  «,  in  the 
plane  of  xz^  and  an  arbitrary  velocity  n  is  imparted  about  this 
axis  alone;  then  v^  and  Vy  will,  at  that  instant,  be  zero,  6=a^ 
and  the  substitution  of  these  values  in  equations  (a)  and  (c)  will 
determine  the  values  of  the  constants  I  and  h. 

h=  —  2  Mff  Y  cos  a 

1=:  On  cos  a, 

which  substituted  in  the  above  equations,  make  them 

.        dip     Cn 
sm^  d --—=z  — -  (cos  o—  cos  a ) 

These  together  with  the  last  equation  (2)  which  may  be  writ- 
ten, (substituting  the  value  of  v^) 

dcpz=zndt-{- cos  0 dip  (5.) 

will,  (if  integrated)  determine  the  three  angles  (t,  ^  and  ^p  in 
terms  of  the  time  t  They  are  therefore  the  differential  equa- 
tions of  motion  of  the  gyroscope. 


J.   G.  BARNARD    ON   THE    GYROSCOPE.  543 

Let  NEE'  (fig.  1)  be  a  section  of  tlie  solid  by  the  plane  ic,  y, . 
This  section  may  be  called  the  equator.  E  being  some  fixed 
point  in  the  equator  (through  which  the  principal  axis  Ox, 
passes),  the  angle  9  is  the  angle  EON. 

If  -A^  is  the  ascending  node  of  the  equator — that  is,  the  point 
at  which  E  in  its  axial  rotation  rises  above  the  horizontal  plane, 
the  angle  (p  must  increase  from  N  towards  E — that  is,  d(p  (in 
equation  5)  must  be  positive  and  (as  the  second  term  of  its  value 
is  usually  very  small  compared  to  the  first)  the  angular  velocity 
n  must  be  positive.  That  being  the  case  the  value  oidq>  will  be 
exactly  that  due  to  the  constant  axial  rotation  ndt^  augmented 
by  the  term  cos  OdH^^  which  is  the  projection  on  the  plane  of  the 
equator  of  the  angular  motion  dip  of  the  node.  This  term  is  an 
increment  to  ndt  when  it  is  positive,  and  the  reverse  when  it  is 
negative.  In  the  first  case,  the  motion  of  the  node  is  considered 
retrograde — in  the  second,  direct. 

The  first  member  of  the  second  equation  (4)  being  essentially 
positive,  the  difference  cos  ^— cos «  must  be  always  positive— 
that  is,  the  axis  of  figure  Oz ,  can  never  rise  above  its  initial  an- 
gle of  elevation  «.     As  a  consequence  -7-  [in  first  equation  (4)] 

must  be  always  positive.  The  node  N,  therefore,  moves  always 
in  the  direction  in  which  V  is  laid  off  positively,  and  the  motion 
will  be  direct  or  retrograde,  with  reference  to  the  axial  rotation, 
according  as  cos^  is  negative  or  positive — that  is,  as  the  axis 
of  figure  is  above  or  below  the  horizontal  plane.  In  either  case 
the  motion  of  the  node  in  its  own  horizontal  plane  is  always 
progressive  in  the  same  direction.  If  the  rotation  n  were  re- 
versed, so  would  also  be  the  motion  of  the  node. 

If  this  rotation  n  is  zero,  -7-  must  also  be  zero  and  the  second 
equation  (4)  reduces  at  once  to  the  equation  of  the  compound 
pendulum,  as  it  should.  Eliminating  ~  between  the  two  equa- 
tions (4)  we  get 

.  ^^dd^     2Mgy  ^.\^       C'-^n'^    ,       .  ^-,  ,      n  \ 

sin2  ^  — —  z=z f-^  fsm^  a —  (cos  0 — cos  a)  |  (cos  ^— cos  a). 

The  length  of  the  simple  pendulum  which  would  make  its 

oscillations  in  the  same  time  as  the  body  (if  the  rotary  velocity 

A 
n  were  zero)  is  t^^.*    If  we  call  this  X  and  make  for  simplicity 


*  The  length  of  the  Bimple  pendulum  is  (see  Bartlett's  Mech,,  p.  252)  X=  — — 

The  moment  of  inertia  A=M(kj^  +7*);  hence  -,-7-  =x. 


644  J.  G.  BARNARD  ON  THE  GYROSCOPE, 

— 7:r-  =  -^  the  above  equation  becomes 

sin  2  0^  =z^  [sin  2  ^-  2 192  (cos  ^-cos a)]  (cos  6»-cos a)     (6) 

and  the  first  equation  (4)  becomes 

^\n2  0-^—2^     f  (cos^-cosa).  (7.) 

Equation  (6)  would,  if  integrated,  give  the  value  of  6  in  terms 
of  the  time ;  that  is,  the  inclination  which  the  axis  of  figure 
makes  at  any  moment  with  the  vertical ;  while  eq.  (7)  (after  sub- 
stituting the  ascertained  value  of  0)  would  give  the  value  of  <// 
and  hence  determines  the  progressive  movement  of  the  body 
about  the  vertical  Oz. 

These  equations  in  the  above  general  form,  have  not  been 
integrated  ;'^  nevertheless  they  furnish  the  means  of  obtaining  all 
that  we  desire  with  regard  to  gyroscopic  motion,  and  m  particu- 
lar that  self-sustaining  power,  which  it  is  the  particular  object 
of  our  analysis  to  explain. 

In  the  first  place,  from  eq.  (6),  by  putting  —  equal  to  zero,  we 

can  obtain  the  maximum  and  minimum  values  of  6.  This  diff. 
co-efficient  is  zero,  when  the  factor  cos  <9— cos  a=:0,  that  is,  when 
<9=a;  and  this  is  sl  maximum,  for  it  has  just  been  shown  from 
equations  (4)  that  q  cannot  exceed  «.  It  will  be  zero  also  and  0 
a  minimum,^  when 

sin2 19— 2i?2  (cos  6  -  cos  a)z=0 
or  cos  6=1  - (92 -f  Vl  -f-2  p~cos  a-ff^.  (g.) 

(The  positive  sign  of  the  radical  alone  applies  to  the  case,  since 
the  negative  one  would  make  0  a  greater  angle  than  «.) 

It  is  clear  that  («  being  given)  the  value  of  0  depends  on  (9 
alone,  and  that  it  can  never  become  zero  unless  ^  is  zero ;  and 
as  long  as  the  impressed  rotary  velocity  n  is  not  itself  zero  (how- 
ever minute  it  may  be),  (5  will  have  a  finite  value. 
It  Thus,  however  rninute  niay  be^_the  velocity  of  rotation,  it  is 
sufiicient  to  prevenFthe  axis  of  rotation  from  faUin(f  to  a  vertical 
I  position. 

The  self-sustaining  power  of  the  gyroscope  when  very  great 
velocities  are  given  is  but  an  extreme  case  of  this  law.  For,  if  § 
is  very  great,  the  small  quantity  1— cos  ^  a  maybe  subtracted 
from  the  quantity  under  the  radical  (eq.  8)  without  sensibly 
idtering  its  value,  which  would  cause  that  eq.  to  become 
cos  0  z=  cos  a. 

*  The  integration  may  be  effected  by  the  use  of  elliptic  functions:  but  the  pro- 
cees  is  of  no  interest  in  this  discussion. 

f  It  is  easy  to  show  that  this  value  of  6  belongs  to  an  actual  minimum ;  but  it  is 
f?carcely  worth  while  to  introduce  the  proof. 


J.  G.  BARNARD  ON  THE  GYROSCOPE.  545 

That  is,  when  the  impressed  velocity  n,  and  in  consequence  ^ 
is  very  great,  the  minimum  value  of  6  differs  from  its  maximum 
a  by  an  exceedingly  minute  quantity. 

Here  then  is  the  result,  analytically  found,  which  so  surprises 
the  observer,  and  for  which  an  explanation  has  been  so  much 
sought  and  so  variously  given.     The  revolvin^y  body,  though  m     ^ 
solicited  by  gravity,  does  not  visibly  fall.  1 1  \^^'-^^'^5> 

Knowing  this  fact,  we  may  assume  that  the  impressed  velocity 
n  is  very  great,  and  hence  cos  <9— cos  «  exceedingly  minute,  and 
on  this  supposition,  obtain  integrals  of  equations  (6)  and  (7), 
which  will  express  with  all  requisite  accuracy  the  true  gyroscopic 
motion.     For  this  purpose,  make 

6=ia — M,  ddzzi—du 

in  which  the  new  variable  u  is  always  extremely  minute,  and  is 
the  angular  descent  of  the  axis  of  figure  below  its  initial  eleva- 
tion. 

By  developing  and  neglecting  the  powers  of  u  superior  to  the 
square,  we  have 

siu^  6  =. sin2  a—  w  sin  2a -|- u^  cos  2a  * 
cos  ^— cos  oczizusmoc^^u^  cos  a 
substituting  these  values  in  eq.  6  we  get 


J 


"dt- 7 .    f 

I  V2wsina  — t^2^(>QS«-[-4/^2j       i 


|9  having  been  assumed  very  great,  cos  a  may  be  neglected  in 
comparison  with  4:^^^^  and  the  above  may  be  written 


J 


I  y^    ^f —    ^  /    T\ 

V  2w  sin  a —•  4/52^2  V  ) 


Integrating  and  observing  that  u  =  o,  when  t  =  o,  we  have 

*  By  Stirling's  theorem, 

in  which  IT,  U\  TJ"  &c.  are  the  values  of/  (m)  and  its  diflferent  co-efficients  when  u 
is  made  zero. 

Making/ (w)  =sin2  (»—«),  and  recollecting  that  sin  lu  =2  sin  u  cos u  and  cos  2w= 
cos*w  —  sin2  M,  we  get  the  value  of  sin^  0  ;  and  making /(w).=  cos(a— w)-cosa 
the  value  in  text  of  cos  B  —  cos  a  is  obtained. 

f  Eq.  6  may  be  written 

•K  d9^         ,  (cos  Q  -  cos  a)2 

-^=2(cos0-cosa)-4/3^  ,i„.  ^      \ 

By  substituting  the  values  just  found,  of  dB,    sin^  Q  and   cos  9  —  cos  a  and  per- 
forming the  operations  indicated,  neglecting  the  higher  powers  of  ii,  (by  which 

fcOS  ^  — "  COS  Ot)  I 

'^Q reduces  simply  to  w^)  and  deducing  the  value     ^  (Z<,  the  expres- 
sion in  the  text,  is  obtained. 

No.  9.— [Vol.  Ill,  No.  2.]— 35. 


546  J.  G.   BARNARD    ON   THE    GYROSCOPE. 

17  ,     1       r       ,    4.s^"l. 

^^.^=-.arc|-cos=l--^_J. 

or,  (since  cos  2a  =  1—2  sin^  a) 

?*  =  ^sinasin2^     l^-.^j  (9.) 

Putting  a— t*  in  place  of  0  (equat.  7)  neglecting  square  of  u,  we  get 

from  which,  observing  that  V  =  0,  when  ^=0 

These  three  expressions  (9),  (10),  (11),  represent  the  vertical 
angular  depression — the  horizontal  angular  velocity — and  the 


sin  a  . 


du  ,  .      ,     ^  2/3         4«2 

„       /-  may  be  put  m  the  form  -: —  . 

*   V2m  sin  a -4/32^2         ''        ^  sma 


J       sin  a 


sin  a 
Call  -T^  =R,  and  the  integral  of  the  2d  factor  of  the  above  is  the  arc  whose  radius 

is  R  and  versed  sine  is  u ;  or  whose  cosine  is  R  —  ^i ;  or  it  is  R  times  the  arc  whose 

u 
cosine  1  —  ^  with  radius  unity.     Substituting  the  value  of  R  in  the  integral  and 

23  1"^ 

multiplying  by  the  factor  7 —  we  get  the  value  of      -^  t,  of  the  text. 

f  In  eq.  (7)  if  we  divide  both  members  by  sin^  9,  and,  in  reducing  the  fraction 

cos  9  -  cos  a 

r---r —  >  use  the  values  already  found  and  neglect  the  square,  as  well  as  higher 

powers  u,  (which  may  be  done  without  sensible  error  owing  to  the  minuteness  of  w, 
though  it  could  not  be  done  in  the  foregoing  values  of  dt  and  t,  since  the  co-efficient 
4^2  in  those  values,  is  reciprocally  great,  as  u  is  small)  the  quotient  will  be  simply 

u 
sin  a 

Substituting  the  value  of  u  and  dividing  out  sin  a  w^e  get  the  value  of  —  in 

the  text. 


The  integral  of  sin  2 13    |  5^  <  <ft  results  from  the  formula    /*  sin2  (p<;q)  =  l(p — 

1   .  11 

-  am 2(p,  easily  obtained  by  substituting  for  sin 2  <p,  its  value  -—-  cos 2(p 


J.   O.   BARN-A 

extent  of  horizontal  angu 
any  time  <.* 

The  first  two  will  reach 

when  sinM-r-^=l   and  = 

These  values  of  t  in  equati' ' 
n 

y=ifr    ■ 

Hence,  counting  from  the 

-r^  and  V  are  all  zero,  we 

a  t 

ponding  values  of  these  var  ^ 

which  correspond  to  the  moi: 
and  -TT  are  maxima,  and 


when,  it  appears  {u  being  th 
gained  its  original  elevatioi 
destroyed. 

All  these  values  are  (owin  . 
very  minute.     If  we  suppose 
100  revolutions  per  second,  t 
ment  of  ordinary  proportion 
of  arc,  and  the  period  of  unc 

Hence  the  horizontal  moti»  \ 
be  exceedingly  slow  compare( 
expressed  by  n. 

If,  in  equations  (9)  and  (10 
find  but  a  repetition  of  the  s 
being  recurring  functions  of  | 

We  see  then  the  revolvin  '; 
uniform  unchanging  elevatio 
port  at  a  uniform  rate,  (as  it  .. 
ure  generates  what  may  be 

*  The  assumption  that  ■vt  =  0  whe 
the  node  coincides  with  the  fixed  a 
analysis  I  suppose  the  initial  positioi 

to  the  above  value  of  -vl,  the  const 

motion  of  the  axis  of  figure  is  the  sj» 


GYROSCOPE. 


ilating  curve  (fig.  2)  wliose 
are  cusps  lying  in  the  same 


h"  b'" 

tch.c'  h',  &c.,  are  to  the  ampli- 
sin  «  :  71.    If  the  initial  ele- 

iiameter  to  the  circumference  of 

.es  the  cycloid. 

3quations  (9)  and  (10)  will  give, 


dt=^'^ 


we  get 

udu 

d  generated  by  the  circle  whose 


)th  the  angles  u  and  v  are  arcs 
)oint  of  the  axis  of  figure  at  a 
to  their  minuteness  may  be  con- 

=:--r-sin  a;    but  then,   while 

the  arc  described  by  the  same 
f  a  small  circle^  whose  actual 
in  the  ratio  of  1 :  sin  a.  The 
circumstances ;  and  the  axis  of 
led  to  the  circumference  of  a 

—  sin  «,   which  rolled   along 

the  vertical  through  the  point 

moves  with  uniform  velocity, 
quation  11)  is  due  to  this  uni- 
mean  'precession. 


J.G.Baxnard  on  the  Gyroscope.  547 

extent  of  horizontal  angular  motion  of  the  axis  of 
figure  after  any  time  t.* 

The  first  two  will  reach  their  respective  maxima  and 
minima  when  ir    fl, 

sin^J^t«4  and  =0|  or  v^ien  "^^"r^Vq  ^^^ 

These  values  of  t  in  equation  (11 )  give 

Hence,  counting  from  the  commencement  of  motion  \f\^en 
t,  u,  J^  and  ^  are  all  zero,  we  have  the  following 
series  of  corresponding  values  of  these  variables 

which  correspond  to  the  moment  of  greatest  depressiont 
when  u  and  ^jf.    are  maxima,  and 

when,  it  appears  (u  being  the  zero),  the  axis  of  figure 
has  regained  its  original  elevation  and  the  horizontal 
velocity  is  destroyed. 

All  these  vcilues  are  (owing  to  the  assumed  large  value 
of /$  )  very  minute.   If  we  suppose  the  rotating  velocity 
n=»100ir  or  100  revolutions  per  second,  the  maximum  of  H 
(with  an  instrument  of  ordinary  proportions)  would  be 
a  fraction  of  a  minute  of  arc,  and  the  period  of  undu- 
lation but  a  fraction  of  a  second. 

Hence  the  horizontal  motion  about  the  point  of  support 
will  be  exceedingly  slow  compared  v;ith  the  axial  rota- 
tion of  the  disk  expressed  by  a.. 

If,  in  equations  (9)  and  (lO),  we  increase  t  indefi- 
nitely, we  will  find  but  a  repetition  of  the  series  of 
values  already  found,  they  being  recurring  functions  of 
the  time. 


'^^^■L^:-■     :.):^^n 


•r      VU 


->    ,:-'f.e:i 


iXi 


»^  ,' 


riPiV: 


-^  X\ 


^^^:j'    : -A 


M"^-' 


547  contd. 

v/e  see  then  the  revolving  body  does  not  in  fact  main- 
tain a  uniform  unchanging  elevation,  and  move  about  its 
point  of  support  at  a  uniform  rate,  (as  it  appears  to  do) 
But  the  axis  of  figure  generates  what  may  be  called 
s-  corrugated  cone ,  and  any 


*  The  assumption  that-y^-O  iriien  t.  is  zero  supposes  that 
the  initial  position  of  the  node  coincides  with  the 
fixed  axis  of  x.   In  my  subsequent  illustrations  and 
analysis  I  suppose  the  initial  position  to  be  at  90° 
therefrom,  which  would  require  to  the  above  value  of 

vA,  the  constant  -'TT  to  be  added.   The  horizontal 
^  Z 

angular  motion  of  the  axis  of  figure  is  the  same  as 
that  of  the  node. 


548 


J. G. Barnard  on  the  Gyroscope 


point  it  would  describe  an  undulating  curve  (fig«2) 
whose  superior  culminations  a,  a  ,  a  ,  &o, ,  are 
CUSPS  lying  in  the  same  horizontal  plane,  eind  whose 

Fig.  2 


sagittae  cb,  ^y   ,  &c. ,  are  to  the  amplitudes  aajfl.a* 

If  the  initial  elevations  is  90^,  this  ratio  is  as 
the  diajneter  to  the  circumference  of  the  circle  :  a 
property  v/hich  indicates  the  cycloid* 

Assuming X  »90°  and  sin«  =1,  equations  (9)  and  (10 ) 

will  give,  by  elimination  of  sin^^  ^^  f. 


dLi- 


-x^ 


x/3;i«. 


substituting  this  value  in  eq. ( d)  we  get 

the  differential  equation  of  the  cycloid  generated  by 
the  circle  whose  diameter  is— 4—  - 

In  this  position  of  the  axis,  both  the  angles  u  and 
1/^  are  arcs  of  great  circles  described  by  a  point  of 
the  axis  of  figure  at  a  units  distance  from  0^,  and 
owing  to  their  minuteness  may  be  considered  as 
rectilinear  co-ordinates. 

If  c{   is  not  900,  the  sagittae  b£=/7isin<\;  but  then-, 


while  the  angular  motion  is  the  same,  the  arc 
described  by  the  same  point  of  the  axis  v/ill  be  that 


tVA  "'  A  v^ 


',*  • !  ":j":  7  ■  T'.i ;  "^^^'^  ;"77 ™r ' 

.  --i  J  J-  o . ;  -c  i  '  0-    "  'C/  •  i>ri  *:  C  x  J" ! 


548  contd. 


of  a  small  circle «  whose  actual  length  will  likewise 
be  reduced  in  the  ratio  of  1  :  since*    The  curve  is 
therefore  a  cycloid  in  all  circumstances;  and  the 
axis  of  figure  moves  as  if  it  were  attached  to  the 
circumference  of  a  minute  circle  whose  dieimeter  is 


•r^  sin  ^ ,  which  rolled  along  the  horizontal  circle, 

aa^a','  about  the  verticeuL  throughthe  point  of  support. 

The  centre  je  of  this  little  circle  moves  with 
uniform  velocity.   The  first  term  of  the  value  of  y 
(equation  ll)  is  due  to  this  uniform  motion  :   it 
may  be  called  the  mean  precession* 


i^o 


:.'^''Ci:o    .>5:.  •-.;  .;..: 


J-.  G.  BARNARD  ON  THE  GYROSCOPE.  549 

The  second  term  is  due  to  the  circular  motion  of  the  axis 
about  this  centre,  and,  combined  with  the  corresponding  values 
of  u^  constitutes  what  may  be  called  the  nutation. 

These  cycloidal  undulations  are  so  minute — succeed  each  other 
with  such  rapidity,  (with  the  high  degrees  of  velocity  usually 
given  to  the  gyroscope,)  that  they  are  entirely  lost  to  the  eye, 
and  the  axis  seems  to  maintain  an  unvarying  elevation  and  move 
around  the  vertical  with  a  uniform  slow  motion. 

It  is  in  omitting  to  take  into  account  these  minute  undulations 
that  nearly  all  popular  explanations  fail.  They  fail,  in  the  first 
place,  because  they  substitute,  in  the  place  of  the  real  phenome- 
non, one  which  is  purely  imaginary  and  inexplicable^  since  it  is 
in  direct  variance  with  fact  and  the  laws  of  nature ; — and  they 
fail,  because  these  undulations — (great  or  small,  according  as  the 
impressed  rotation  is  small  or  great)  furnish  the  only  true  clue 
to  an  understanding  of  the  subject. 

^The  fact  is,  that  the  phenomenon  exhibited  by  the  gyroscope 
which*  is  so  striking,  and  for  which  explanations  are  so  much 
sought,  is  only  a  particular  and  extreme  phase  of  the  motion  ex- 
pressed by  equations  (6)  and  (7) — that  the  self-sustaining  power 
is  not  absolute^  but  one  of  degree — ^that  however  minute  the  axial 
rotation  may  be,  the  body  never  will  fall  quite  to  the  vertical ; — 
however  great,  it  cannot  sustain  itself  without  any  depression. 

I  have  exhibited  the  undulations,  as  they  exist  with  high  veloci- 
ties,— when  they  become  minute  and  nearly  true  cycloids  ;  with 
low  velocities,  they  would  occupy  (horizontally)  a  larger  portion 
of  the  arc  of  a  semi-circle,  and  reach  downward  approximating, 
more  or  less  nearly,  to  contact  with  the  vertical :  and,  finally^ 
when  the  rotary  velocity  is  zero,  their  cusps  are  in  diametrically 
opposite  points  of  the  horizontal  circle,  while  the  curves  resolve 
themselves  into  vertical  circular  arcs  which  coincide  with  each 
other,  and  the  vibration  of  the  pendulum  is  exhibited.  All 
these  varieties  of  motion,  of  which  that  of  the  pendulum  is  one 
extreme  phase  and  the  gyroscopic  another,  are  embraced  in 
equations  (6)  and  (7)  and  exhibited  by  varying  §  from  0  to  high 
values,  though,  (wanting  general  integrals  to  these  equations) 
we  cannot  determine,  except  in  these  extreme  cases,  the  exact 
-  elements  of  the  undulations.  The  minimum  value  of  0  may 
however  always  be  determined  by  equation  (8). 

If  we  scrutinize  the  meaning  of  equations  (6)  and  (7),  it  will 
be  found  that  they  represent,  the  first,  the  horizontal  angular 
component  of  the  velocity  of  a  point  at  units  distance  from  0, 
and  the  second,  the  actual  velocity  gf  such  point.* 

*  In  more  general  terms  equations  (4)  express,  the  first,  that  the  moment  of  the 
quantity  of  motion  about  the  fixed  vertical  axis  Oz  remains  always  constant :  the 
second  that  the  living  forces  generated  in  the  body  (over  and  above  the  impressed 
axial  rotation)  are  exactly  what  is  due  to  gravity  through  the  height,  h. 

Both  are  expressions  of  truths  that  might  have  been  anticipated ;  for  gravity 


660  J.  G.  BARNARD  ON  THE  GYROSCOPE. 

For  sin  ^  -77  is  the  horizontal,  and  -7-  the  vertical,  component 

of  this  velocity.     Calling  the  first  t'A,  and  the  second  v^,  and 
the  resultant  Vg^  and  calling  cos  ^— cos  «,  (which  is  the  true 
height  of  fall)  A,  those  equations  may  be  written 
Cn     h 
''=  A   ^6  <^) 

This  velocity  Vs  (as  a  function  of  the  height  of  fall)  is  exactly 
that  of  the  compound  pendulum,  and  is  entirely  independent  of  the 
axial  rotation  n.  Hence,  (as  we  might  reasonably  suppose)  ro- 
tary motion  has  no  power  to  impair  the  work  of  gravity  through 
a  given  height,  in  generating  velocity ;  but  it  does  have  power  to 
change  tJie  direction  of  that  velocity.  Its  effect  is  precisely  that  of 
a  material  undulatory  curve,  which,  deflecting  the  body's  path 
from  vertical  descent,  finally  directs  it  upward,  and  causes  its 
velocity  to  be  destroyed  by  the  same  forces  which  generated  it. 

And  it  may  be  remarked,  that,  were  the  cycloid,  we  have  de- 
scribed, ^wc/i  a  material  curve,  on  which  the  axis  of  the  gyroscope 
rested,  without  friction  and  without  rotation,  it  would  travel  along 
this  curve  by  the  effect  of  gravity  alone,  (the  velocity  of  descent 
on  the  downward  branch  carrying  it  up  the  ascending  one,)  with 
exactly  the  same  velocity  that  the  rotating  disk  does,  through  the 
combined  effects  of  gravity  and  rotation. 

Equation  {a)  expresses  the  horizontal  velocity  produced  by 
the  rotation. 

If  we  substitute  its  value  in  the  second,  we  may  deduce 


de__  \2g        C-^n^     h^ 


If  we  take  this  value  at  the  commencement  of  descent,  and 
before  any  horizontal  velocity  is  acquired,  (making  h  indefinitely 
small),  the  second  term  under  the  radical  may  be  neglected,  and 

the  first  increment  of  descending  velocity  becomes  ^  h,  pre- 
cisely what  is  due  to  gravity,  and  what  it  would  he  were  there  no 
rotation. 

Hence  the  popular  idea  that  a  rotating  body  offers  any  direct 
resistance  to  a  change  of  its  plane,  is  unfounded.  It  requires  as 
little  exertion  of  force  (in  the  direction  of  motion)  to  move  it 

cannot  increase  the  moment  of  the  quantity  of  motion  about  an  axis  parallel  to 
itself;  while  its  power  of  generating  living  force  by  working  through  a  given 
height,  cannot  be  impaired. 

Had  we  considered  ourselves  at  liberty  to  assume  them,  however,  the  equations 
might  have  been  got  without  the  tedious  analysis  by  which  we  have  reached  them. 


J.  G.  BARNARD  ON  THE  GYROSCOPE.  551 

from  one  plane  to  another,  as  if  no  rotation  existed ;  and  (as  a 
corollary)  as  little  expenditure  of  work. 

But  deflecting  forces  are  developed,  by  angular  motion  given 
to  the  axis,  and  normal  to  its  direction,  which  are  very  sensible, 
and  are  mistaken  for  direct  resistances.  If  the  extremity  of  the 
axis  of  rotation  were  confined  in  a  vertical  circular  groove,  in 
which  it  could  move  without  friction ;  or  if  any  similar  fixed  re- 
sistance, as  a  material  vertical  plane,  were  opposed  to  the  de- 
flecting force,  the  rotating  disk  would  vibrate  in  the  vertical 
plane,  as  if  no  rotation  existed.    Its  equation  of  motion  would 

become  that  of  the  compound  pendulum,  — -=    V^h,     What 

then  is  the  resistance  to  a  change  of  plane  of  rotation  so  often 
alluded  to  and  described'/     A  mzinomS^lentirely. 

The  above  may  be  otherwise  establisEeJ  Ifin  equations  (3) 
we  introduce  in  the  second  member  an  indeterminate  horizontal 
force,  g\  applied  to  the  centre  of  gravity,  parallel  to  the  fixed 
axis  of  ?/,  and  contrary  to  the  direction  in  which,  in  our  figure, 
we  suppose  the  angle  v  to  increase,  the  projections  of  this  force 
on  the  axes  Ox^^  Oy^^  will  be  a'g'  and  h' g'  and  the  last  two  of 
these  equations  will  become,  (calhng  cosines  x^Oy  and  y^Oy^ 
a'  and  6',) 

Advy—{C—A)nVxdt^^yM(ag-ira'g')dt 
AdVa:+\C—A)nVydt=-yM(bg-{-h'g')dt 
Multiplying  the  first  by  Vy  and  the  second  by  v^  and  adding 
A  (vydvy  -^Vxdvx  )=zyM\g  (a  Vy^hvj;)d  t-\-g'  (a'  Vy—h'vj;)d  t~\. 

But  (avy—bva;)dt}iSiS  been  shown  (p.  53)  to  be  =d.cosO^ — and 
by  a  similar  process  it  may  be  shown  that  {a'Vy  —  'b'Vx)dt= 
=d.  (sin  0  cos  V').     (For  values  of  a'  and  h\  see  p.  52.) 

Let  us  suppose  now  that  the  force  g'  is  such  that  the  axis  of 
the  disk  may  be  always  maintained  in  the  plane  of  its  initial  po- 
sition xz.  The  angle  v^  would  always  be  90°,  dip=0,  and  c?.(sin^ 
cos  y^)=0.  That  is,  the  co-efficient  of  the  new  force  g'  becomes 
zero;  and  the  integral  of  the  above  equation  is  as  before  (p.  54), 
A(Vy2+Va;^)=2YMg  cosd-^h.    , 

But  the  value  of  Vy^+v^^  likewise  reduces  (since  -^=0)  to  -7-^ 

and  the  above  becomes  the  equation  of  the  compound  pendulum. 

dO^     2Yj}fg  2,  a 

(g)         7772^^ — A —  ^^^  ^"^~^^^X  ^^^^  ^— cos  a),  (h  being  determined.) 

This  is  the  principle  just  before  announced,  that,  with  a  force  so 
applied  as  to  prevent  any  deflection  from  the  plane  in  which 
gravity  tends  to  cause  the  axis  to  vibrate,  the  motion  would  be 
precisely  as  if  no  axial  rotation  existed. 


552  J.  G.  BARNARD  ON  THE  GYROSCOPE. 

To  determine  the  force  of  g' ;  multiply  tlie  first  of  preceding 
equations  by  J,  and  the  second  by  a,  and  add  the  two,  and  add 
likewise  A{vydh+Vxda)=—AndQOQd  (see  p.  54)  and  we  shall 
get 

Ad{hVy-\-aVx)-^Cndco^d=:yMg'{a'h  —  ah')dt. 
By  referring  to  the  values  of  a,  a',  h,  h\  and  performing  the 
operations  indicated  and  making  cos  ^=o,  sin  v=l,  the  above 
becomes, 

Ad{bvy-\-aVa:)-\-Ond  cos  0=YMg^  sin  6 dt. 

But  the  value  of  {hvy+av:^)  (p.  54)  becomes  zero  when  --^^=0. 

TT  /      OndcosO  Cn  dd  ^ 

Hence  g  :=—% ^^ — = —  * 

yM&mOdt         yMdt 

The  second  factor  —  is  the  angular  velocity  with  which  the  axis 

of  rotation  is  moving. 

Hence  calling  Vs  that  angular  velocity,  the  value  of  the  deflect- 
ing force^  g'  may  be  written  (irrespective  of  signs), 

^-^^^-  (^) 

that  is,  it  is  directly  proportional  to  the  axial  rotation  n,  and  to 
the  angular  velocity  of  the  axis  of  that  rotation.  By  putting  for 
0,  Mk^  (in  which  h  is  the  distance  from  the  axis  at  which  the 
mass  M^  if  concentrated,  would  have  the  moment  of  inertia,  (7,) 
the  above  takes  the  simple  form 

In  the  case  we  have  been  considering  above,  in  which  g'  is  sup- 
posed to  counteract  the  deflecting  force  of  axial  rotation,  the  angu- 
lar velocity  Vs  ,ot—j-  (equation  g)  is  equal  to     hr  (^^^  ^  ~  ^^^  ")• 

But  in  the  case  of  the  free  motion  of  the  gyroscope,  this  de- 
flecting force  combines  with  gravity  to  produce  the  observed 
movements  of  the  axis  of  figure. 

If,  therefore,  we  disregard  the  axial  rotation  and  consider  the 
body  simply  as  fixed  at  the  point  0,  and  acted  upon,  at  the  cen- 
ter of  gravity,  by  two  forces — one  of  gravity,  constant  in  inten- 
sity and  direction — the  other,  the  deflecting  force  due  to  an  axial 

C 
rotational,  whose  variable  intensity  is  represented  hj—=rz.nvsj 

*  The  effect  of  gravity  is  to  diminish  9  and  the  increment  dd  is  negative  in  the 
case  we  are  considering.  Hence  the  negative  sign  to  the  value  of  ^',  indicating  that 
the  force  is  in  the  direction  of  the  positive  axis  of  y,  as  it  should,  since  the  tendency 
of  the  node  is  to  move  in  the  reverse  direction. 


J.  G.  BARNARD  ON  THE  GYROSCOPE.  553 

and  whose  direction  is  always  normal  to  the  plane  of  motion  of 
the  axis ;  we  ought,  introducing  these  forces,  and  making  the 
axial  rotation  n  zero,  in  general  equations  (8),  to  be  able  to  de- 
duce therefrom  the  identical  equations  (4)  which  express  the  mo- 
tion of  the  gyroscope.  •  a 

This  I  have  done ;  but  as  it  is  only  a  Verification  of  what  has 
previously  been  said,  I  omit  in  the  text  the  introduction  of  the 
somewhat  difficult  analysis.* 

Equation  (5)  becomes  (in  the  case  we  consider),  by  integration, 

(p■=^nt^\^^J  cos  a 

which,  with  the  values  of  u  and  V  already  obtained,  determines 
completely  the  position  of  the  body  at  any  instant  of  time. 

Knowing  now  not  only  the  exact  nature  of  the  motion  of  the 
gyroscope,  but  the  direction  and  intensity  of  the  forces  which 

*  To  introduce  these  forces  in  eq,  (3)  I  observe,  first,  that  as  both  are  applied  at 
O  (in  the  axis  Oz^  the  moment  L^  is  still  zero  and  the  first  eq.  becomes,  as  before, 
CdVg  =  0  or  Vg=:  const. 
And  as  we  disregard  the  impressed  axial  rotation,  we  make  this  constant  (or  v^  ) 
zero. 

Cn 
The  deflecting  force  — ^  Vg  (taken  with  contrary  sign  to  the  counteracting  force 

Cn  d9  Cn  d-^ 

just  obtained)  resolves  itself  into  two  components  — t>  -jt  and  —  —^  -jr  sin  9,  the 

first  in  a  horizontal,  the  second  in  a  vertical  plane,  and  both  normal  to  the  axis  of 
figure. 

The  second  is  opposed  to  gravity,  whose  component  normal  to  the  axis  of  figure, 
is  g  sin  9. 

Hence  we  have  the  two  component  forces  (in  the  directions  above  indicated), 
^  Cnd^        ,        /       Cn  c?^^  \ 

'Wf'dl       ^iff--^7irr-^]  sine. 


/       Cnd^\    . 

^[^-WdF)  ''' 

I       Cn  dA,\     , 
,^  /        Cn  d-^\     ,    ^       .        ^^Cn  d^ 


These  moments  with  reference  to  the  axes  of  y  j  and  x  j  will  be 
.    "      ^  /        Cn  dA,\     ,    ^  ,^  C'n   (f9 

-sm(P7Jf  \9-;7M-ir\  8me-cos(P7if  ^  ^,  and 


Hence  equations  (3)  (making  v^  zero,  and  putting  for  M^  and  Ni  the  above  values, 
and  recollecting  the  values  of  a  and  b,  (p.  53)  become 

d-^  d^      1 

Advy  =  a<yMgdt  —  aCn  -^  dt—  Cn  cos  ^~i7dt 


KdVj.  =—  bjMgdt-\-bCn  -^  dt—  Cn  sin  (p  ^  <ft 


(0 


Multiplying  the  equations  severally  by  Vy  and  v^,  adding  and  reducing  (as  on 
p.  53)  we  get 

A{vydvy\-  Vj.dVj;)=z  jMgd  .cos^  —  Cn-yrd.  cos  9—  Cn  d^  (  Vy  cos  (p+v*  sin  (p) 
But  Vy  cos  (p-f-Va;  sin  (p  will  be  found  equal  to  sin  9  -jt  (by  substituting  the  values 


554  J.  G.  BARNARD  ON  THE  GYROSCOPE. 

produce  it,  it  is  not  difficult  to  understand  why  such  a  motion 
takes  place. 

Fig.  1  represents  the  body  as  supported  by  a  point  within  its 
mass ;  but  the  analysis  applies  to  any  position,  in  the  axis  of 
figure,  within  or  without ;  and  figs.  3  and  4  represent  the  more 
familiar  circumstances  under  which  the  phenomenon  is  ex- 
hibited. 

Let  the  revolving  body  be  supposed  (fig.  3,  vertical  projection), 
for  simplicity  of  projection,  an  exact  sphere^  supported  by  a 
point  in  the  axis  prolonged,  at  0,  which  has  an  initial  elevation 
a  greater  than  90°.  Fig.  4  represents  the  projection  on  the  hor- 
izontal plane  xy\  the  initial  position  of  the  axis  of  figure  (being 
in  the  plane  of  xz)  is  projected  in  Ox. 

Ox^  Oy^  Oz^  are  the  three  (fixed  in  space)  co-ordinate  axes,  to 
which  the  body's  position  is  referred. 

In  this  position,  an  initial  and  high  velocity  n  is  supposed  to 
be  given  about  the  axis  of  figure  Os  j ,  so  that  the  visible  por- 
tions move  in  the  direction  of  the  arrows  5,  h\  and  the  body  is 
left  subject  to  whatever  motion  about  its  point  of  support  0, 
gravity  may  impress  upon  it.  Had  it  no  axial  rotation,  it  would 
immediately  fall  and  vibrate  according  to  the  known  laws  of  the 
pendulum.  Instead  of  which,  while  the  axis  maintains  (appar- 
ently) its  elevation  «,  it  moves  slowly  around  the  vertical  Oz,  re- 
ceding from  the  observer,  or  from  the  position  ON"  towards  ON. 

It  is  self-evident  that  the  first  tendency  (and  as  I  have  likewise 
proved,  the  first  effect)  of  gravity  is  to  cause  the  axis  Oz^  to  de- 
scend vertically,  and  to  generate  vertical  angular  velocity.  But 
with  this  angular  velocity,  the  deflecting  force  proportional  to 
that  velocity  and  normal  to  its  direction,  is  generated,  which 
pushes  aside  the  descending  axis  from  its  vertical  path. — But  as 
the  direction  of  motion  changes,  so  does  the  direction  of  this 
force — always  preserving  its  perpendicularity.   It  finally  acquires 

of  vy  and  v^) ;  hence  the  two  last  terms  destroy  each  other,  and  the  above  equation 
becomes  identical  with  equation  (a)  from  which  the  2d  eq.  (4)  is  deduced. 

Multiplying  the  1st  equation  \i)  by  coscp  and  the  second  by  sin(p  and  adding, 
we  get, 

^(cos  <pdvy  -f-  sin  (pdv^)  =  -  Cnd  9. 

By  differentiating  the  values  of  Vy  and  v^,  performing  the  multiphcations,  and 
substituting  for  d(p  its  value,  cos  9  d-^,  (proceeding  from  the  3d  equation  (2)  when 
?;^=0)  the  above  becomes 

/         J2^l.  d-^  d^\  ^   d9 

^  ['^ « -dl^  +2 '^^ s  'at  ^)=-(^^ w 

Multiplying  both  members  by  sin  0  dt,  and  integrating,  the  above  becomes 
,        d-^        Cn 

the  same  as  the  1st  equation  (4)  when  the  value  of  the  constant  I  is  determined. 


^  J^cJ^  -  p, 


^^6^ 


J.  G.  BARNARD  ON  THE  GYROSCOPE. 


655 


an  intensity  and  upward  direction  adequate  to  neutralize  the 
downward  action  of  gravity ;  but  the  acquired  downward  velocity 
still  exists  and  the  axis  still  descends  at  the  same  time  acquiring 
a  constantly  increasing  horizontal  component,  and  with  it  a  still 
increasing  upward  deflecting  force.     At  length  the  descending 

Fig.  3. 


&^ 


cuiiipv^nent  of  velocity  is  entirely  destroyed — the  path  ot  the 
axis  is  horizontal ;  the  deflecting  force  due  to  it  acts  directly 
contrary  to  gravity,  which  it  exceeds  in  intensity,  and  hence 
causes  the  axis  to  commence  rising.  This  is  .the  state  of  things 
at  the  point  h  (fig.  2).   The  axis  has  descended  the  curve  a  b,  and 


556 


J.    G.   BARNARD    ON   THE    GYROSCOPE. 


has  acquired  a  velocity  due  to  its  actual  fall  a  c?;  but  this  velocity 
has  been  deflected  to  a  horizontal  direction.  The  ascent  of  the 
branch  h  a'  is  precisely  the  converse  of  its  descent.  The  acquired 
horizontal  velocity  impels  the  axis  horizontally,  while  the  de- 
flecting force  due  to  it  (now  at  its  maximum)  causes  it  to  com- 
mence ascending.  As  the  curve  bends  upward,  the  normal 
direction  of  this  force  opposes  itself  more  and  more  to  the  hori- 
zontal, while  gravity  is  equally  counteracting  the  vertical,  veloc- 
ity. As  the  horizontal  velocity  at  h  was  due  to  a  fall  through  the 
height  a  c?,  so,  through  the  medium  of  this  deflecting  force,  it  is 
just  capable  of  restoring  the  work  gravity  had  expended  and 
lifting  the  axis  back  to  its  original  elevation  at  a',  and  the  cy- 
cloidal  undulation  is  completed,  to  be  again  and  again  repeated, 
and  the  axis  of  figure,  performing  undulations  too  rapid  and  too 
minute  to  be  perceived,  moves  slowly  around  its  point  of  sup- 
port. 

Eeferring  to  fig.  8,  the  equator  of  the  revolving  body  (a  plane 
perpendicular  to  the  axis  of  figure  and  through  the  fixed  point  0,) 
would  be  an  imaginary  plane  E^  E^.  Its  intersection  with  the 
horizontal  plane  oi  xy  would  be  the  line  of  nodes  Aj  JSf'.  In 
the  position  delineated,  the  progression  of  the  nodes  is  direct. 
For,  at  the  ascending  node  A,  any  point  in  the  imaginary  plane 
of  the  equator  (supposed  to  revolve  with  the  body)  would  move 
upwards  in  the  direction  of  the  arrow  a,  while  the  node  moves 
in  the  same  direction  from  0  (of  the  arrow  a').    Were  the  axis  of 

Fig.  5. 


figure  below  the  horizontal  plane,  (fig.  5)  the  upward  rotation  of 
the  point  would  be  from  0  to  E^  (as  the  arrow  a),  while  the  pro- 
gression of  the  node  (in  the  same  direction  as  before  as  the  arrow 
a')  would  be  the  reverse,  and  the  motion  of  the  node  would  be 
retrograde — yet  in  both  cases  the  same  in  space. 


J.  G.  BARNARD  ON  THE  GYROSCOPE. 


557 


Fig.  6. 


As  the  deflecting  force  of  rotary  motion  is  tlie  sole  agent  in 
diverting  the  vertical  velocity  produced  by  gravity  from  its 
downward  direction,  and  in  producing  these  paradoxical  effects ; 
and  as  the  foregoing  analysis  while  it  has  determined  its  value, 
has  thrown  no  light  upon  its  origin,  it  may  be  well  to  inquire 
how  this  force  is  created. 

Popular  explanations  have  usually  turned  upon  the  deflexion 
of  the  vertical  components  of  rotary  velocity  by  the  vertical  an- 
gular motion  of  the  axis  produced  by  gravity.  In  point  of  fact, 
however,  both  vertical  and  horizontal  components  are  deflected, 
one  as  much  as  the  other ;  and  the  simplest  way  of  studying  the 
effects  produced,  is  to  trace  a  vertical  projection  of  the  path  of  a 
point  of  the  body  under  these  combined  motions.  For  this  pur- 
pose conceive  the  mass  of  the  revolving  disk  concentrated  in  a 
single  ring  of  matter  of  a  radius  h  due  to  its  moment  of  inertia 
C=Mh^^  (see  Bartlett  Mech.  p.  178)  and,  for  simplicity,  suppose 
the  angular  motion  of  the  axis  to  take  place  around  the  centre 
figure  and  of  gravity  G. 

Let  AB  \)Q  such  a  ring 
(supposed  perpendicular  to 
the  plane  of  projection)  re- 
volving about  its  axis  of  fig- 
ure G  (7,  while  the  axis  turns 
in  the  vertical  plane  about  the 
same  point  Q.  Let  the  rota- 
tion be  such  that  the  visible 
portion  of  the  disk  moves 
upward  through  the  semi-cir- 
cumference, from  B  io  A^ 
while  the  axis  moves  down- 
ward through  the  angle  d  to 
the  position  G  C.  The  point 
-5,  by  its  axial  rotation  alone, 

would  be  carried  to  A ;  but  the  plane  of  the  disk,  by  simultane- 
ous movement  of  the  axis,  is  carried  to  the  position  A' B'  and 
the  point  ^arrives  at  B'  instead  of  A,  through  the  curve  pro- 
jected mBGB'  The  equation  of  the  projection,  in  circular 
lunctions,  is  easily  made;  but  its  general  character  is  readilv 
perceived,  and  it  is  sufficient  to  say,  that  it  passes  through  the 
^Tl;  tT  i*^  tangents  at  ^ and  B  are  perpendicular  to  ^^  " 
f^  A  -f 'T^^4  r^*  i^s  concavity,  throughout  its  whole  length, 
turned  to  the  right  The  point  A  descends  on  the  other,  or  re- 
mote side  of  the  disk,  and  makes  an  exactly  similar  curve  AG  A' 
with  its  concavity  reversed. 

The  centrifugal  forces  due  to  the  deflections  of  the  vertical 
motions  are  normal  to  the  concavities  of  these  curves ;  hence,  on 
the  side  of  the  axis  towards  the  ejp,  they  are  to  the  left,  and  on 


558  J.  G.  BARNARD    ON   THE    GYROSCOPE. 

the  opposite  or  further  side,  to  the  right^  (as  the  arrows  h  and  a.) 
Hence  the  joint  effect  is  to  press  the  axis  G  (7  from  its  vertical 
plane  GGG'^  horizontally  and  towards  the  eye.  Eeverse  the  di- 
rection of  axial  rotation  and  the  curves  A  A'  and  BB  will  be 
the  same,  except  that  A  A'  would  be  on  the  wear,  and  BB  on 
the  remote  side  of  the  axis  G  (7,  and  the  direction  of  the  result- 
ing pressure  will  be  reversed. 

A  projection  on  the  horizontal  plane  would  likewise  illustrate 
this  deflecting  force  and  show  at  the  same  time  that  there  is  no 
resistance  in  the  plane  of  motion  of  the  axis,  and  that  the  whole 
effect  of  these  deflexions  of  the  paths  of  the  different  material 
points,  is  a  mere  interchange  of  living  forces  between  the  different 
material  points  of  the  disk  ;  but  it  is  believed  that  the  foregoing 
illustration  is  sufficient  to  explain  the  origin  of  this  force,  whose 
measure  and  direction  I  have  analytically  demonstrated. 

It  may  be  remarked,  however,  that  the  intensity  of  the  force 
will  evidently  be  directly  as  the  velocities  gained  and  lost  in  the 
motion  of  the  particles  from  one  side  of  the  axis  to  the  other ; 
or  as  the  angular  velocity  of  the  axis,  and  as  the  distance,  Jc,  of  the 
particles  from  that  axis.  It  will  also  be  as  the  number  of  particles 
which  undergo  this  gain  and  loss  of  living  force  in  a  given  time ; 
or  as  the  velocity  of  axial  rotation.  Considered  as  applied  nor- 
mally at  G  to  produce  rotation  about  any  fixed  point  0  in  the 
axis,  its  intensity  will  evidently  be  directly  as  the  arm  of  lever  k, 
and  inversely  as  the  distance  of  G  from  0  (/).     Hence  the  meas- 

ure  of  this  force  already  found,  from  analysis,  g'=  — nvs. 

In  the  foregoing  analysis,  the  entire  ponderable  mass  is  sup- 
posed to  partake  of  the  impressed  rotation  about  the  axis  of  fig- 
ure Oz  1 ;  and  such  must  be  the  case,  in  order  that  the  results  we 
have  arrived  at  may  rigidly  apply.  Such,  however,  cannot  be 
the  case  in  practice.  A  portion  of  the  instrument  must  consist 
of  mountings  which  do  not  share  in  the  rotation  of  the  disk. 
It  is  believed  the  analysis  will  apply  to  this  case  by  simply  in- 
cluding the  whole  mass,  in  computing  the  moment  of  inertia  A 
and  the  mass  M,  while  the  moment  G  represents,  as  before,  that 
of  the  dish  alone. 

In  this  manner  it  would  be  easy  to  calculate  what  amount  of 
extraneous  weight  (with  an  assumed  maximum  depression  u),  the 
instrument  would  sustain,  with  a  given  velocity  of  rotation. 

The  analogy  between  the  minute  motions  of  the  gyroscope 
and  that  grand  phenomenon  exhibited  in  the  heavens, — the 
"  precession  of  the  equinoxes" — is  often  remarked.  In  an  ulti- 
mate analysis,  the  phenomena,  doubtless,  are  identical ;  yet  the 
immediate  causes  of  the  latter  are  so  much  more  complex,  that 
it  is  difficult  to  institute  any  profitable  comparison. 


J.  G.  BARNARD  ON  THE  GYROSCOPE.  559 

At  first  sight,  the  undulatory  motion  attending  the  precession, 
known  as  "  nutation"  (nodding)  would  seem  identical  with  the 
undulations  of  the  gyroscope.  But  the  identity  is  not  easily  indi- 
cated ;  for  the  earth's  motion  of  nutation  is  mainly  governed  by  the 
moon,  with  whose  cycles  it  coincides ;  and  the  solar  and  lunar 
precessions  and  nutations  are  so  combined,  and  affected  by  causes 
which  do  not  enter  into  our  problem,  that  it  is  vain  to  attempt 
any  minute  identification  of  the  phenomena,  without  reference 
to  the  difficult  analysis  of  celestial  mechanics. 

On  a  preceding  page,  I  said  that  a  horizontal  motion  of  the 
rotating  disk  around  its  point  of  support,  without  descending 
undulations,  was  at  variance  with  the  laws  of  nature.  This  as- 
sertion applied  however  only  to  the  actual  problem  in  hand,  in 
which  no  other  external  force  than  gravity  was  considered,  and 
no  other  initial  velocity  than  that  of  axial  rotation. 

Analysis  shows,  however,  that  an  initial  impulse  may  be  ap- 
plied to  the  rotating  disk  in  such  a  way  that  the  horizontal  mo- 
tion shall  be  absolutely  without  undulation.  An  initial  horizon- 
tal angular  velocity  such  as  would  make  its  corresponding  de- 
flective force  equal  to  the  component  of  gravity,  g  sin  (9,  would 
cause  a  horizontal  motion  without  undulation. 

If  the  axial  rotation  n^  as  well  as  the  horizontal  rotation,  is 
communicated  by  an  impulsive  force,  analysis  shows  that  it  may 
be  applied  in  any  plane  intersecting  the  horizontal  plane  in  the 
line  of  nodes  ;  but  if  applied  in  the  plane  of  the  equator  (where 
it  can  communicate  nothing  but  an  axial  rotation  n),  or  in  the 
horizontal  plane,  its  intensity  must  be  infinite. 

My  announced  object  does  not  carry  me  further  into  the  con- 
sideration of  the  gyroscope  than  the  solution  of  this  peculiar 
phenomenon,  which  depends  solely  upon,  and  is  so  illustrative 
of,  the  laws  of  rotary  motion. 

If  I  have  been  at  all  successful  in  making  this  so  often  ex- 
plained subject  more  intelligible — in  giving  clearer  views  of  some 
of  the  supposed  effects  of  rotation,  it  has  been  because  I  have 
trusted  solely  to  the  only  safe  guide  in  the  complicated  phenom- 
ena of  nature,  analysis, 

[The  foregoing  analysis  of  the  phenomana  of  the  Gyroscope,  by  Major  Barnard, 
of  the  Corps  of  Engineers  of  the  United  States  Army,  and  late  Superintendent  of 
the  Military  Academy  at  West  Point,  is  inserted  in  this  Journal,  although  it  will  also 
appear  in  the  "■American  Journal  of  Science  and  Art"  for  July,  because  many  of 
our  readers  have  become  interested  in  the  subject  from  the  articles  which  have 
already  appeared  in  our  pages,  and  because  we  have  been  asked  for  a  more  scientific 
explanation  of  what  has  been  called  the  self-sustaining  power  in  the  rotary  disc  \IL 
The  length  of  the  paper  has  crowded  many  articles  of  educational  intelligence  into  ' '  • 
the  next  number.    Ed.] 


560 


THE    GTKOSCOPE. 


c3^^</. 


The  following  patterns  of  Gyroscopes  can  be  safely  sent  by  Express  to  any  one 
remitting  the  price : — 

No.  1.  Gyroscope  of  Iron,  simplest  form, $2.00 

"     2.  "        "  Brass      "        " 3.00 

3.  "        "      "    sphere  in  place  of  wheel,      .    .     .      3.50 

4.  "         "       "     with  lever  and  weight, 4.00 

5.  "         "       "     with  socket  and  arms, 5.00 

6.  "         "       "     hung  on  gymbals, 7.00 

7.  "        "      "    with  three  concentric  rings,  (next  page,)  8.00 

8.  "        "  Rings  of  brass  with  lever  &  weight  large  size,  10.00 
Nos.  5,  7,  and  8,  are  b^t  for  Schools  and  Colleges. 

Address  at  Hartford,  Conn.,  F.  C.  Brownell,  Secy. 

"        "  Chicago,  111.,  Talcott  &  Sherwood. 


v/«-^«>*^^  ^e^  /ssy^,  ^^ 


XVI.    EDUCATIONAL   MISCELLANY. 

ON  THE  MOTION  OF  THE  GYROSCOPE  AS  MODIFIED  BY  THE  RETARDING 
FORCES  OF  FRICTION,  AND  THE  RESISTANCE  OF  THE  AIR*. 

WITH  A  BRIEF  ANALYSIS  OF  THE  TOP.    O^^^c^lJi^C  " 

BY  MAJOE  J.   G.  BAENAED,  A.  M. 
Corps  of  Engineers  U.  S.  A. 


In  a  previous  paper  (see  article  in  this  Joumal  for  June, 
1857,  to  wMcli  this  paper  is  intended  to  be  supplementary,) 
I  have  investigated  the  ''Self-sustaining  power  of  the  Gyro- 
scope" in  the  light  of  analysis.  From  the  general  equations 
of  "Eotary  motion"  I  have  deduced  the  laws  of  motion  for 
the  particular  case  of  a  solid  of  revolution  moving  about  a  fixed 
point  in  its  axis  of  figure,  (or  the  prolongation  thereof).  I 
have  shown  that  such  a  body,  having  its  axis  placed  in  any 
degree  of  inclination  to  the  vertical,  and  having  a  high  rotary 
motion  about  that  axis,  will  not,  under  the  influence  of  grav- 
ity, sensibly  fall ;  but  that  any  point  in  the.  axis  will  describe 
"an  undulating  curve  whose  superior  culminations  are  cusps 
lying  in  the  same  horizontal  plane ;"  that  this  curve  approaches 
more  and  more  nearly  to  the  cycloid,  as  the  velocity  of  axial 
rotation  is  greater ;  that  when  this  velocity  is  very  great  the 
undulations  become  very  minute  and  "  the  axis  of  figure  per- 
forming undulations  too  rapid  and  too  minute  to  be  perceived, 
moves  slowly  about  its  point  of  support."  I  have  shown  how 
the  direction  and  velocity  of  this  gyration  are  determined  by  the 
direction  and  velocity  of  axial  rotation  and  the  distance  of  the 
center  of  gravity  of  the  figure  from  the  point  of  support,  and 
that  the  remarkable  phenomenon  exhibited  by  the  gyroscope  is 
but  a  particular  case  due  to  a  very  high  velocity  of  axial  rotation, 
of  the  general  laws  of  motion  of  such  a  body  as  described, 
which  embrace  the  motion  of  the  pendulum  in  one  extreme  and 
that  of  the  gyroscope  in  the  other,  and  that  intermediate  between 
these  two  extreme  cases  (for  moderate  rotary  velocities)  the  un- 
dulations of  the  axis,  will  be  large  and  sensible. 

I  have  likewise  shown  that  whenever,  to  the  axis  of  a  rotating 
solid,  an  angular  velocity  is  imparted,  a  force  which  I  have 
called  "  the  deflecting  force^"*  acting  perpendicular  to  the  plane  of 
motion  of  that  axis,  is  developed,  whose  intensity  is  proportional, 
to  this  angular  velocity,  and  likewise  to  the  rotary  velocity  of 
the  body ;  and  that  it  is  this  deflecting  force  which  is  the  imme- 
diate sustaining  agent,  in  the  gyroscope. 

In  the  above  deductions  of  analysis  is  found  the  full  and  com- 
plete solution  of  the  "  self-sustaining  power  of  the  gyroscope." 

To  make  the  character  of  the  motion  indicated  by  analysis, 

No.  11.— [IV.,  No.  2.]— 34. 


530  ^-  ^'  BARNARD  ON  THE  GYROSCOPE. 

sensible  to  the  eye,  it  is  only  necessary  to  attach  to  the  ordinary 
gyroscope,  in  the  prolongation  of  the  axis,  an  arm  of  five  or  six 
inches  in  length,  and  having  an  universal  joint  at  its  extremity, 
and  to  swing  the  instrument  as  a  pendulum ;  or,  the  extremity 
of  an  arm  of  such  a  length  may  be  rested  in  the  usual  way, 
upon  the  point  of  the  standard,  when,  with  the  centre  of  gyra- 
tion removed  at  so  great  a  distance  from  the  point  of  support, 
the  undulatory  motion  becomes  very  evident. 

But  it  cannot  fail  to  be  observed  that  the  motion  preserves 
this  peculiar  feature  but  for  a  very  short  period.  The  undula- 
tions speedily  disappear;  instead  of  periodical  moments  of  rest 
(which  the  theory  requires  at  each  cusp)  the  gyratory  velocity 
becomes  continuous^  and  nearly  uniform  and  horizontal;  audit 
increases  as  the  axis  (owing  to  the  retarding  influences  of  friction 
and  the  resistance  of  the  air)  slowly  falls.  In  short,  the  axis 
soon  seems  to  move  upon  a  descending  spiral  described  about  a 
vertical  through  the  point  of  support. 

The  experimental  gyroscope,  in  its  simplest  form  consists  of 
two  distinct  masses,  the  rotating  disk,  and  the  mounting  (or  ring 
in  which  the  disk  turns).  The  point  of  support  in  the  latter, 
though  it  gives  free  motion  about  a  vertical  axis,  constrains 
more  or  less,  the  motion  of  the  combined  mass  about  any  other. 
The  rotating  disk  turns  at  the  extremities  of  its  axle,  upon 
points  or  surfaces  in  the  mass  of  the  mounting,  with  friction ;  it 
is  rare,  too,  that  the  point  of  support,  of  the  mounting,  is  ad- 
justed in  the  exact  prolongation  of  the  axis  of  the  disk. 

Without  attempting  to  subject  to  analysis  causes  so  difficult 
to  grasp  as  these,  I  shall  first  attempt  to  show,  by  general  con- 
siderations, what  would  be  the  immediate  influence  of  the  re- 
tarding forces  of  friction  and  the  resistance  of  the  air  upon  our 
theoretical  solid ;  and  then  point  out  the  further  effect  due  to  the 
discrepancies  of  figure,  above  indicated.  Leaving  out  of  con- 
sideration the  minute  effect  of  friction  at  the  point  of  support, 
these  forces  exert  their  influence,  mainly  in  retarding  the  rotary 
velocity  of  the  disk.  Friction — at  the  extremities  of  the  axle  of 
the  disk,  and  the  resistance  of  the  air,  at  its  surface,  are  power- 
ful enough  to  destroy  entirely  in  a  Yerj  few  minutes,  the  high 
velocity  originally  given  to  it.  It  is  in  this  way,  mainly,  that 
they  modify  the  motion  indicated  by  analysis. 

If  the  rotary  velocity  remained  co?25^ar2  Awhile  the  axis  made  one 
of  the  little  cycloidal  curves  aba',  (fig.  1)  the  deflecting  force 
would  be  just  sufficient,  as  I  have  shown  (p.  556  of  the  article 
cited)  to  lift  the  axis  back  to  its  original  elevation  a',  and  to 
destroy,  entirely,  the  velocity  it  had  acquired  through  its  fall  cb. 
If,  at  a',  the  rotary  velocity  n  underwent  an  instantaneous  dimi- 
nution, and  remained  constant  through  another  undulation,  a 
curve,  of  larger  amplitude  and  sagitta  a'  b'  a"  would  be  described, 
and  the  axis  would  again  rise  to  its  original  elevation  a",  and 
again  be  brought  to  rest.   We  might  then,  on  casual    considera- 


J,  G.  BARNARD  ON  THE  GYROSCOPK 


531 


tion  of  the  subject,  expect  to  see  the  undula- 
tions become  more  and  more  sensible  as  the 
rotary  velocity  decreased.  The  reverse,  how- 
ever, is  the  case,  as  I  have  already  stated.  In 
fact,  the  above  supposition  would  require  the 
rotary  velocity  n  to  be  a  discontinuous  decreas- 
ing function  of  the  time ;  whereas  it  is,  really 
a  continuous  decreasing  function.  It  is  under- 
going a  gradual  diminution  between  a  and  a'. 
The  deflecting  force,  which  is  constantly  pro- 
portional to  it,  is  therefore  insufficient  to  keep 
the  axis  up  to  the  theoretical  curve  aha',  but 
a  bluer  Guive  ah^a^  is  described;  and  when 
the  culmination  a ,  is  reached,  it  is  helow  the 
original  elevation  a\ 

But  the  2d  of  our  general  equations  for  the 
gyroscope  (4),  [afterwards  put  under  the  sim- 
ple  form  jeq.  (f)\vs^  =—h'\  which  is  inde- 

pendent  of  n,  shows  that  the  angular  velocity 
of  the  axis  will  always  be  that  due  to  its  actual 
fall  h  below  the  initial  elevation.  On  reaching 
the  culmination  a ,  therefore,  the  axis  will  not 
come  to  rest,  but  will  have  a  horizontal  veloc- 
ity due  to  the  fall  a'a^  and  the  curve  will  not 
form  a  cusp  but  an  inflexion  at  a^. 

The  axis  will  commence  its  second  descent, 
therefore,  with  an  initial  horizontal  velocity. 
It  will  not  descend  as  much  as  it  would  have 
done  had  it  started  from  rest  with  its  dimin- 
ished value  of  n ;  and,  for  the  same  reason 
as  before,  will  not  be  able  as  again  to  rise 
high  as  its  starting  point  a^  but  to  a  some- 
what lower  point  a^  and  with  an  increased 
horizontal  velocity.  These  increments  of  hori- 
zontal velocity  will  constantly  ensue  as  the 
culminations  become  lower  and  lower,  while 
on  the  other  hand,  the  undulations  become  less 
and  less  marked,  as  indicated  by  the  ligure. 

I  have  stated  in  my  former  paper  (p.  559) 
that  a  certain  initial  horizontal  angular  velocity 
such  as  would  "  make  its  corresponding  deflect- 
ing force  equal  to  the  component  of  gravity,  g 
sin  <5,  would  cause  a  horizontal  motion  without  undulation."  This 
horizontal  velocity  is  rapidly  attained  through  the  agencies  just 
described :  or,  at  least,  nearly  approximated  to,  and  the  axis,  as 
observation  shows,  soon  acquires  a  continuous  and  uniform  hori- 
zontal motion. 

On  the  other  hand,  this  sustaining  power  being  directly  pro- 


532  J-  ^-  BARNARD  OJM  THE  GYROSCOPE. 

portional  to  the  rotary  velocity  of  the  disk,  as  well  as  to  the  an- 
gular velocity  of  the  axis,  diminishes  with  the  former,  and  as  it 
diminishes,  the  axis  must  descend,  acquiring  angular  velocity  due 
to  the  height  of  fall :  hence  the  rapid  gyration  and  the  descend- 
ing spiral  motion  which  accompanies  the  loss  of  rotary  velocity. 

A  more  curious  and  puzzling  effect  of  the  friction  of  the  axle 
is  presented,  when  we  come  to  take  into  consideration,  instead 
of  our  theoretical  solid,  the  discrepancies  of  figure  presented  by 
the  actual  gyroscope.  If,  with  a  high  initial  rotation,  the  com- 
mon gyroscope  be  placed  on  its  point  of  support  with  its  axis 
somewhat  inclined  above  a  horizontal  position,  it  will  soon  be 
observed  to  rise.  In  my  analytical  examination  (p.  543)  I  have 
stated  as  a  deduction  from  the  second  equation  (4),  that  "  the 
axis  of  figure  can  never  rise  above  its  initial  angle  of  elevation." 
That  equation  supposes  that  the  rotary  velocity  n  remains  unim- 
paired, and  is  the  expression  of  a  fundamental  principle  of  dy- 
namics— that  of  "living  forces"  (so-called),  which  requires  that 
the  living  force  generated  by  gravity  be  directly  proportional  to 
the  height  of  fall,  and  involves  as  a  corollary  that  through  the 
agency  of  its  own  gravity  alone,  the  centre  of  gravity  of  a  body 
can  never  rise  above  its  initial  height.*  The  anomaly  observed, 
therefore,  either  requires  the  action  of  some  foreign  force ;  or^ 
that  the  living  force  lost  by  the  rotating  disk,  shall,  through 
some  hidden  agency,  be  expended  in  performing  this  work  of 
lifting  the  mass. 

The  discrepancy  here  exhibited  between  the  motion  proper  to 
our  theoretical  solid  of  revolution  and  the  experimental  gyro- 
scope is  due  to  the  division  of  the  latter  into  two  distinct  masses, 
one  of  which  rotates,  loith  friction,  upon  points  or  surfaces  in  the 
other ;  and  to  the  fact  that  at  the  point  of  support  (in  the  latter) 
there  is  r^oi  perfectly  free  motion  in  all  directions. 

The  friction  at  the  extremities  of  the  axle  of  the  disk,  tends 
to  impress  on  the  mass  which  constitutes  the  "mounting,"  a  ro- 
tation in  the  same  direction.  Were  the  motion  of  the  latter 
upon  its  fixed  point  of  support  perfectly  free,  the  mounting  and 
disk  would  soon  acquire  a  common  rotatory  velocity  about  the 
axis  of  the  disk.  But  the  mounting  is  perfectly  free  to  turn 
about  the  vertical  axis  through  the  point  of  support,  though  not 
about  any  other.  If  we  decompose,  therefore,  the  rotation  which 
would  be  impressed  upon  the  mounting  into  two  components, 
one  about  this  vertical,  and  the  other  about  a  horizontal  axis — 
the  first  X2k.QQfull  effect,  and  the  latter  is  destroyed  at  the  point 
fo  support.  If  the  axis  of  the  instrument  is  above  the  horizon- 
tal, this  component  of  rotation  is  in  the  same  direction  as  the 
gyration  due  to  gravity,  and  adds  to  it ;  if  the  axis  is  below  the 
horizontal,  the  component  is  the  reverse  of  the  natural  gyration, 
and  diminishes  it. 

*  The  first  of  these  equations  (as  I  have  remarked  in  a  note  to  p.  547)  is  the  expres- 
sion of  another  fundamental  principle — more  usually  called  the  "  principle  of  areas." 


J.  G.  BARNARD  ON  THE  GYROSCOPE.  533 

But  I  have  shown  that  the  axis  soon  acquires,  independent  of 
this  cause,  a  gyration  whose  deflecting  or  sustaining  force  is  just 
equivalent  to  the  downward  component  of  gravity.  The  addi- 
tion to  this  gyratory  velocity  caused  by  friction  when  the  axis  is 
inclined  upwards  puts  the  deflecting  force  in  eoccess^  and  the  axis 
is  raised ;  it  is  raised,  as  in  all  other  cases  in  which  work  is  done 
through  acquired  velocity — viz.,  by  an  expenditure  of  living 
force ;  but  in  this  instance,  through  a  most  curious  and  compli- 
cated series  of  agencies. 

The  phenomenon  may  be  best  illustrated  in  the  following  man- 
ner. Let  the  outer  extremity  of  the  common  gyroscope,  having 
its  axis  inclined  above  the  horizontal,  be  supported  by  a  thread 
attached  to  some  fixed  point  vertically  above  the  point  of  support, 
so  that  gyration  shall  be  free.  Here  gravity  is  eliminated,  and 
the  axis  of  our  theoretical  solid  of  revolution  would  remain  per- 
fectly motionless ;  but  the  gyroscope  starts  off,  of  itself,  to  gy- 
rate in  the  same  direction  that  it  would  were  its  extremity  ^ree. 
This  gyration  increases  (if  the  rotary  velocity  is  great)  until  the 
deflecting  force  due  to  it,  lifts  the  outer  extremity  from  its  sup- 
port on  the  thread,  and  it  continues  indefinitely  to  rise.  Try 
the  same  experiment  with  the  axis  helow  the  horizontal.  The 
gyration  will  commence  spontaneously  as  before,  but  in  the 
reverse  direction :  it  will  increase  until  the  inner  extremity  is  lifted 
from  the  point  of  support^  (the  action  of  the  deflecting  force  being 
here  reversed,)  the  instrument  supporting  itself  on  the  thread 
alone.  If  the  experiment  is  tried  with  the  axis  perfectly  hori- 
zontal, no  gyration  takes  place,  for  the  component  of  rotation, 
due  to  friction,  is,  in  this  position,  zero. 

The  foregoing  reasoning  accounts,  I  believe,  for  all  the  ob- 
served phenomena  of  the  experimental  gyroscope,  and  shows 
how,  from  the  theory  of  oar  imaginary  solid  of  revolution,  a 
consideration  of  the  effects  of  the  discrepancies  of  form,  and  of 
the  actual  disturbing  forces,  leads  to  their  satisfactory  explanation. 

The  great  similarity  between  the  phenomena  of  the  top  and 
gyroscope,  renders  it  not  uninteresting  to  compare  the  laws  of 
motion  of  the  two.  If  we  conceive  a  solid  of  revolution  ter- 
minated at  its  lower  extremity  by  &  point  (the  ordinary  form  of 
the  top),  resting  upon  a  horizontal  plane  without  fi-iction,  and 
having  a  rotary  motion  about  its  axis  of  figure,  such  a  body  will 
be  subject  to  the  action  of  two  forces;  its  weight,  acting  at  the 
centre  of  gravity,  and  the  resistance  of  the  plane,  acting  at  the 
point  vertically  upwards. 

According  to  the  fundamental  principles  of  dynamics,  the 
centre  of  gravity  will  move  as  if  the  mass  and  forces  were  con- 
centrated at  that  point,  while  the  mass  will  turn  about  this  cen- 
tre as  if  it  were  fixed.  Calling  E  the  resistance  of  the  plane, 
if  the  mass,  and  Mg  the  weight  of  the  top,  and  z  the  height  of 


634 


J.  G.  BARNARD  ON  THE  GYROSCOPE. 


the  centre  of  gravity  above  the  plane,  we  shall  have  for  the 
equation  of  motion  of  the  centre  of  gravity* 

^^£=^-^^         (!•) 

As  the  angular  motion  of  the  body  is  the  same  as  if  the  centre 
of  gravity  was  fixed,  and  as  R  is  the  only  force  which  operates 
to  produce  rotation  about  that  centre,  if  we  call  G  the  moment 
of  inertia  of  the  top  about  its  axis  of  figure,  and  A  its  moment 
with  reference  to  a  perpendicular  axis  through  the  centre  of 
gravity,  and  /  the  distance,  GK  (fig.  2)  of  the  point  of  support 
from  that  centre;  the  equations  of  rotary  motion  will  become 
identical  with  equations  (3)  (p.  541),  substituting  B  for  Mg 
Cdv^=0  ') 

Advy-^{C—A)v^Vxdt  =  yaRdt      >      (2.) 
Advx-\-{C^A)vyVzdt=:—yhRdt  ) 
The  first  of  equations  (2)  gives  us  Vz  as  for  the  gyroscope, 
equal  a  constant  n. 

Multiplying  the  2d  and  8d  of  equations  (2)  by  Vy  and  v^  re- 
spectively, and  adding  and  making  the  same  reduction  as  on  p. 
63,  we  shall  get 

A{yydvy-\-Vxdvx)=:Ry  d  ,co^d. 

But  z  (the  height  of  the  centre  of  gravity  above  the  fixed  plane) 
=  —  y cos (9 ;  hence  yd.Q>o&d  =—dz;  and  equation  (1)  gives 

-r-^ -h-g ).     Substituting  these  values  of  E and  yd.aosO  in 

the  preceding  equation,  and  integrating,  we  have 

A{vy2+Vx2)^M{^^  +  2gz'^=k  (3.) 

From  the  2d  and  3d  of  equations  (2)  the  equation  (c)  (of  the 
gyroscope,  p.  542)  is  deduced  by  an  identical  process. 

A(bVy-\-aVa;)-\-  On  co&  6:=:l, 
and  a  substitution  in  the  two  foregoing  equations  of  the  values 
of  the  cosines  a  and  h,  and  of  the  angular  velocities  v^  and  Vy, 
in  terms  of  the  angles  (p,  0  and  ip  (see  pp.  540,  541),  and  for  z  and 

— -  their  values,  —  /  cos (9,  and  '/sin  (9—  and  a  determination  of  the 
dt  d  t 

constants,  on  the  supposition  of  an  initial  inclination  of  the  axis 
a,  and  of  initial  velocity  of  axial  rotation  w,  will  give  us  for  the 
equations  of  motion  of  the  top : 

hm^d-j-zzz—r-  (cos  (9 — cos  a) 

Ai^^m2e^-^^^J1^ 

*  As  there  are  no  horizontal  forces  in  action,  there  can  be  no  horizontal  motion 
of  the  centre  of  gravity  except  from  initial  impulse,  which  I  here  exclude. 


J.  G.  BARNARD  ON  THE  GYROSCOPE.  535 

from  wliicli  the  angular  motions  of  the  top  can  be  determined. 
The  first  is  identical  with  the  first  equation  (4)  for  the  gyroscope. 
The  second  differs  from  the  jsecond  gyroscopic  equation  only  in 

containing  in  its  first  member  the  term  My^  ^in^d-—^  or  its 

equivalent  M-j-^ ,  expressing  the  living  force  of  vertical  transla- 
tion of  the  whole  mass. 

The  second  member  (as  in  the  corresponding  equation  for  the 
gyroscope)  expresses  the  work  of^  gravity^  and  the  first  term  of 
the  first  member  expresses  the  living  force  due  to  the  angular 
motion  of  the  axis.  Instead  therefore  of  the  work  of  gravity 
being  expended  (as  in  the  gyroscope)  ivhoUy  in  producing  angu- 
lar motion,  part  of  it  is  expended  in  vertical  translation  of  the 
centre  of  gravity.  The  angular  motion  takes  place  not  (as  in 
the  gyroscope)  about  the  point  of  support  (which  in  this  case  is 
not  fixed\  but  about  the  centre  of  gravity  (to  which  the  moments 
of  inertia  A  and  B  refer) ;  and  that  centre,  motionless  horizon- 
tally, moves  vertically  up  and  down,  coincident  with  the  small 
angular  undulations  of  the  axis  through  a  space  which  will  be 
more  and  more  minute  as  the  rotary  velocity  n  is  greater. 

An  elimination  of  -r-  between  the  two  equations  (4)  and  a 

study  of  the  resulting  equation,  would  lead  us  to  the  same  gen- 
eral results,  as  the  similar  process,  p.  544,  for  the  gyroscope. 

The  vertical  angular  motion,  expressed  by  the  variation  which 
the  angle  0  undergoes,  becomes  exceedingly  minute  (the  maxi- 
mum and  minimum  values  of  6  approximating  each  other)  when 
n  is  great,  and  the  axis  gyrates  with  slow  undulatory  motion 
about  a  vertical  through  the  centre  of  gravity.  It  would  be 
easy,  likewise,  to  show  by  substituting  for  0  another  variable, 
u=ct—dj  always  (in  case  of  high  values  of  n)  extremely  small, 
and  whose  higher  powers  may  therefore  be  neglected,  that  the 
co-ordinates  of  angular  motion,  u  and  V,  approximate  more  and 
more  nearly  to  the  relation  expressed  by  the  equation  of  the 
cycloid  as  n  increases ;  though  the  approximation  is  not  so  rapid 
as  in  the  gyroscope.  All  the  results  and  conclusions  flowing 
from  the  similar  process  for  the  gyroscope  (see  pp.  545,  546,  547, 
548)  would  be  deduced.  As,  however,  the  centre  of  gravity,  to 
which  these  angular  motions  are  referred,  is  not  a  fixed  ])oint^ 
but  is  itself  constantly  rising  and  falling  as  d  increases  or  di- 
minishes, the  actual  motion  of  the  axis  is  of  a  more  complicated 
character. 

If  OK"  (see  fig.  2)  is  the  initial  position  of  the  axis  of  the 
top,  the  motion  of  the  centre  of  gravity  will  consist  ina  vertical 
falling  and  rising  through  the  distance  GG'—  GK"{q,o^z^  G'G"— 
coaZiG  G") = y  (cos  ^ ,  —  cos  «)  (in  which  0  ^  is  the  minimum  value  of  ^) 


536 


J.  G.  BARNARD  ON  THE  GYROSCOPE. 


while  the  extremity  of  the  axis 
or  pointy  K^  describes  on  the 
supporting  surface  and  about 
the  projection  G"  of  the  cen- 
tre of  gravity,  an  undulating 
curve  a,  Z>,  a',  h'^  a'\  &c.,  hav- 
ing cusps  a,  a\  ko,.^  in  the  circle 
described  about  G"  with  the 
radius  G"K"—y  sina,  and 
tangent,  externally,  to  the 
circle  described  with  a  radius 
G"  K'=y  sin^,.  But,  as  in 
the  case  of  the  gyroscope, 
these  little  undulations  speedi- 
ly disappear  through  the  re- 
tarding influence  of  friction 
and  resistance  of  the  air,  and 
the  point  of  the  top  describes 
about  G". 


a  circle,  more  or  less  perfect. 


The  rationale  of  the  self-sustaining  power  of  the  top  is  identi- 
cal with  that  of  the  gyroscope ;  the  deflecting  force  due  to  the 
angular  motion  of  the  axis  plays  the  same  part  as  the  sustaining 
agent,  and  has  the  same  analytical  expression.  Owing  io  friction^ 
the  top  likewise  rises,  and  soon  attains  a  vertical  position ;  but 
the  agency  by  which  this  effect  is  produced  is  not  exactly  the 
same  as  for  the  gyroscope. 

If  the  extremity  of  the  top  is  rounded,  or  is  not  a  perfect 
mathematical  point,  it  will  roll^  by  friction,  on  the  supporting 
surface  along  the  circular  track  just  described.  This  rolling 
speedily  imparts  an  angular  motion  to  the  axis  greater  than  the 
horizontal  gyration  due  to  gravity,  and  the  deflecting  force  be- 
comes in  excess,  (as  explained  in  the  case  of  the  gyroscope,)  and 
the  axis  rises  until  the  top  assumes  a  vertical  position.  Even 
though  the  extremity  of  the  top  is  a  very  perfect  point,  yet  if  it 
happens  to  be  slightly  out  of  the  axis  of  figure  (and  rotation)  the 
same  result  will,  in  a  less  degree,  ensue :  for  the  point,  instead 
of  resting  permanenthj  on  the  surface,  will  strike  it,  at  each  revo- 
lution, and  in  so  doing,  propel  the  extremity  along.  The  condi- 
tions of  a  perfect  point,  perfectly  centered  in  the  axis  of  figure, 
are  rarely  combined,  or  rather  ixre  practically  impossible;  but  it  is 
easy  to  ascertain  by  experiment  that  the  more  nearly  they  are 
fulfilled,  and  the  harder  and  more  highly  polished  the  support- 
ing surface,  the  less  tendency  to  rise  is  exhibited ;  while  the 
great  stiffness  (or  tendency  to  assume  a  vertical  position)  of  tops 
with  rounded  points,  is  a  fact  well  known  and  made  use  of  in 
the  construction  of  these  toys. 

C^f"The  references  throughout  this  paper  are  to  my  paper  on  the  gyroscope  in  the 
June  number  of  the  Am.  Journal  of  Education. 


^££^9'^^  ,      /SO^  - 


"99 


XVIII.   EDUCATIONAL  MISCELLANY  AND  INTELLIGENCE. 

0^  THE  EFFECTS  OF  INITIAL  GYRATORY  VELOCITIES,    AND  OF  RETARDDfG  FORCES, 
ON  THE  MOTION  OF  THE  GYROSCOPE. 

BY   MAJOR  J.    G.   BARNARD,   A.    M 

Corps  of  Engineers,  U.  S.  A.* 


In  one  of  the  concluding  paragraphs'  of  my  first  paper  on  the  Gyro- 
scope (Am.  Journal  of  Education,  June,  1857,)  I  stated  that  "  an  initial  im- 
pulse may  be  applied  to  the  rotating  disk  in  such  a  way  that  the  horizon- 
tal motion  shall  be  absolutely  without  undulation.  An  initial  angular 
velocity  such  as  would  make  its  corresponding  deflective  force  equal  to 
the  component  of  gravity  g  sin  ^,  would  cause  a  horizontal  motion  without 
undulation." 

The  statement  contained  in  the  last  sentence  quoted,  is  not  rigidly  true ; 
for  besides  the  component  of  gravity,  there  is  another  force  to  be  consid- 
ered, viz.,  the  centrifugal  force  due  to  the  gyratory  velocity,  which  acts 
either  in  conjunction  with,  or  in  opposition  to,  the  component  of  gravity, 
according  as  the  axis  of  the  disk  is  above  or  below  a  horizontal. 

In  this  last  position  this  force  is  null  (as  regards  its  effects  in  sustaining 
or  depressing  the  axis),  and  to  this  angular  elevation  of  the  axis  the 
statement  quoted  is  true  without  qualification.  The  assumption  of  an 
initial  horizontal  velocity  requires  only  a  new  determination  of  constants 
for  equations  (a)  and  (c)  (pp.  541,  542,  June  No.). 

If  we  make,  in  those  equations 

^=a,  ()d:zi90°,  V=90°,  «=-sina,  v^zzim,  Vy=0,  VzZ=:.n, 
(in  which  m  is  the  assumed  initial  velocity)  and  determine  the  constants 
h  and  I  therefrom,  the  equations  of  motion  will  become 

sm  2  o  —  =z  —  (cos  o  —  cos  a)  -|-  m  sm  a 

.         Bm^^^  +  — =-^(cos6-cos«)  +  m^  J 

and  from  them  we  get 

.   ^^dO^      r^Mgy  .  ^^     "iCmn  .  C^n^ ,      ^ 

sm2  d  — — -=z|  i^-^  sm^  0^ —  sm  a — —  (cos  c^— cos  a) 

dt^      \^    A  A  ^2    V  / 

—  m2  (cos  6 -|- cos  a)  I  (cos  ^— cos  a)     (2) 

dd  dip 

From  this  we  ffet  -;-=  0  when  cos  ^  —  cos  a  =0 :  and  as  -^  is  not  zero 
^     dt  '  dt 

for  this  initial  elevation,  it  indicates,  instead  of  a  cusp,  a  tangency  to  the 

horizontal  here. 

This  paper  is  intended  to  give  a  more  rigidly  mathematical  demonstration  of 
the  effects  of  "  retarding  forces"  than  is  given  in  (December  No.  p.  529,)  of  this  Jour- 
nal >■  and  to  give  the  theory  of  the  "  motions "  of  the  Gyroscope  a  more  general 
form,  by  the  introduction  of  "  Initial  Gyratory  Velocities." 


300 


J.  G.  BARNARD  ON  THE  GYROSCOPE. 


If  the  curve  described  is  horizontal  without  undulation,  the  other  fac- 
tor of  the  second  member  of  eq.  (2)  should  likewise  become  zero  with 
dz=za :  an  effect  which  may  ensue  from  a  suitable  value  given  to  m. 
The  value  of  the  deflecting  force  due  to  a  given  angular  velocity  m  is 
(J 
(p.  552,  June  number)  -—mn,  and  if  we  suppose  this  equal  to  the  com- 
ponent of  gravity  g  sin  «,  we  shall  have  m  j=  --f^sin  a. 

Cn 

If  we  substitute  this  value  of  m  in  the  second  member  of  equation  (2) 
and  assume  a  =r  90°  the  factor  in  question  becomes  zero  for  ^z=  a,  and 
the  maximum  and  minimum  values  of  d  are  the  same,  indicating  a  hori- 
zontal motion  without  undulation. 

For  every  other  initial  elevation  than  90°  a  different  value  of  m  is  re- 
quired to  produce  this  result,  in  consequence  of  the  influence  of  the  cen- 
trifugal force  of  gyration  at  other  elevations. 

With  «=  90°,  equation  (2)  becomes 

'dt^~\    —J-^^'"^^ -J ^^cos<9-.m2cos^     cos(9  (3) 

Placing  the  first  factor  of  the  second  member  equal  to  zero  and  solving 
with  reference  to  cos^  we  get  (recollecting  the  value  given  to  §  in  our 
former  article) 


sin2  e  - 


52-4?-  + 


4:Mgy 


^V^4.Mgy] 


+  1- 


Cmn 
Mgy' 


(4) 


For  ^/^  =  0,  equation  (3)  expresses  the  cycloidal  curve  with  cusps  a,  a', 

a'\  &c.,  as  has  been  already  shown  in  our  former  investigation.     For 

M  g'^ 
m  >  0  but  <^  —^ —  the  minimum  value  of  6  derived  from  equation  (4)  is 

greater  than  when  m  is  zero,  while  instead  of  a  cusp  (there  is  as  has 
already  been  observed)  a  tangency  at  a,  and  the  curve  has  the  wave  form 
a6j  a'h\  (the  points  b^b^'b^",  &c.  being  higher  than  bb'b").* 

Mgy 
When  mr=-— —  the  curve  unites  with  the  horizontal  a  a' a"  a'"  and 
Cn 

there  is  no  undulation ;  equation  (4)  giving  cos  ^  =:  0,  or  ^  =  90°. 


*  In  reality,  the  amplitudes,  a  a',  a'  a",  of  the  undulations  become  increased,  at  the 
same  time  that  the  sagittic  are  diminished,  but,  for  the  sake  of  comparison,  I  have 
represented  them  the  same  for  each  variety  of  curve. 


J.  G.  BAENARD  ON  THE  GYROSCOPE.  301 

When  m  >► --^, -r-  becomes  still  zero  with  ^z=a  =  90°;  but  this 
Cn    at 
instead  of  a  njaximum  is  now  a  minimum  value  of  ^,  for  the  value  of  0 
which  satisfies  equation  (4)  is  greater  than  90°,  and  the  curve  ah^  a'h^', 
&c.,  undulates  above  the  plane  a  a'  a". 

2  Mo  Y  1 

Finally  when  m=     ^      ,  equation  (4)  will  give  cos5=-  -^  and  a 

substitution  of  this  in  the  first  equation  (1)  (making  a  =  90°),  will  give 

— r=  0  :  showing  that  the  curve  makes  cusps  at  its  superior  culminations, 

and  that  the  common  cycloidal  motion  is  resumed.     In  fact  the  value  of 

—-  =  -3  {^  (p.  547,  June  number)  at  the  lowest  point  h  of  the  cycloid,  is, 

2  Mo  y 
(substituting  the  values  of  ^  and  i)  exactly  equal  to   —^ ,  and  the 

value  of  the  sagitta  u  corresponding  to  e 6  is  what  we  have  just  found  for 

cos<9,  or  e 63,  viz.  — -. 

If  now,  retaining  m  constant  at  this  value  to  which  we  have  brought 

it,  we  increase  the  rotary  velocity,  w,  or  vice  versa,  a  curve  with  loops,  (fig. 

2,)  may  be  described,  as  it  can  be  shown  that,  for  the  maximum  value 

d  ip 
of  dj  —  becomes  negative.* 

2. 


In  my  supplementary  paper  in  the  December  number  of  this  Journal  I 
have  endeavored  to  show  how  the  theoretical  cycloidal  motion  of  a  sim- 
ple solid  of  revolution  is  modified  by  the  retarding  forces  of  friction  and 
the  resistance  of  the  air,  and  that  the  theory  explains  all  the  phenomena 
observed  in  the  ordinary  gyroscope. 

It  may  be  objected  however  that  the  nature  of  the  curve  given  in 
Fig.  1,  (p.  531,)  is  in  some  degree  assumed,  and  I  therefore"  wish  to  show 
that  it  can  be  confirmed  by  mathematical  demonstration. 

The  rotary  velocity  n  of  the  disk  is  supposed  to  be  gradually  destroyed 
through  the  retarding  forces  of  friction  at  the  extremities  of  the  axle, 
and  of  the  resistance  of  the  air  at  the  surface. 

Without  attempting  to  give  analytical  expressions  for  the  retarding 
forces,  it  is  sufliicient  to  say  that  the  rotary  velocity,  at  the  end  of  any 

*If  m  is  made  negative  and  small  (i.  e.,  a  backward  initial  velocity  given)  a  looped 
curve  like  the  above,  but  lying  below  the  plane  a  a'  a",  results.  All  these  curves  (n 
being  always  supposed  very  great)  are  but  the  different  forms  of  the  "  cycloid " 
known  as  prolate,  common,  and  curtate  cycloids ;  the  common — a  particular  case  of 
the  curve — corresponding  to  the  particular  case  of  the  problem  in  which  the  initial 

gyratory  velocity  is  either  zero  or  has  the  particular  value  —^ • 


302  J.  G.  BARNARD  ON  THE  GYROSCOPE. 

time  i,  counting  from  the  commencement  of  motion,  may  be  expressed 
thus 

in  which  n  is  the  initial  rotary  velocity  of  the  disk. 

If  we  substitute  this  expression  for  v^  in  the  last  two  equations  (3)  (p. 
541,  June  No.,)  and  follow  a  similar  process  to  that  by  which  equations 
(4)  of  that  paper  are  deduced,  we  shall  get,  for  the  equations  of  motion 

sin2|9— =:— -  (cos(9-cosa)-—  /  f{t)d.co&d     7 

at        A  Af/  0  (  /k\ 

For  the  sake  of  simplicity  suppose  the  initial  position  of  the  axis  be  hori- 
zontal, or  a  z=90  and  the  above  become 

sin2^^  =  ^cos^^-^  rf(t)d.coB,e  ) 

dt        A  Aj  o'^^  (  ,_. 

dt  ^  dt^         A  .  J 


« 
If  aff  a'  represents  the  cycloidal  curve,   and  aee'  e"  g'  the  curve  in 
question,  it  will  be  observed  that  the  angular  velocity  of  the  axis  given 
by  the  2nd  equation  (6)  is  the  same  for  both,  for  equal  values  of  ^,  while 

the  value  of  i^e  horizontal  component  oi  \h2ii  velocity,  sin^— ,  is  less 

0       r^ 

than  for  the  cycloidal  curve,  by  the  term  -  /  f{t)d.  cos  (9. 

A  sm  C  </  Q 

As  6  diminishes,  d  cos  6  is  positive  and  this  term  is  subtractive  and 

dip 
hence  for  any  point  e  or  e'  on  the  descending  branch,  — -  is  less  than  for 

the  corresponding  point /or/'  of  the  cycloid,  and  the  branch  aee'  e"  will 
be  behind  the  branch  aff,  and  will  descend  lower. 

C        P 
At  e"  the  term  — j— ; — -r-  /   flt)d.  cos  d.  attains  its  maximum,  for  as  the 
A&md  Jo 

curve  ascends,  0  increases,  and  the  increments  of  cos  0  become  negative. 

*  When  the  retarding  force  is  independent  of  the  velocity,  as  in  the  case  of  fric- 
tion, the/(^)  in  the  above  expression  is  linear;  when  this  force  is  dependent  upon 
the  velocity,  as  for  the  resistance  of  the  air,/(^)  will,  in  general,  be  an  infinite  and 
diverging  series  in  the  powers  of  t ;  whether  the  force  is  due  to  either,  or  both 
combined,  of  these  causes,  the  above  expression  for  the  velocity  of  rotation  may 
howe\spr  be  used  for  the  present  purpose. 


J.  G.  BARNARD  ON  THE  GYROSCOPE.  303 

But  as  the  values  of  t  on  this  branch  of  the  curve  are  nearly  double  those 
or  equal  values  of  0  of  the  descending  one,  the  integral  /  f(t)d.cosO 
will  become  zero  at  some  point  ff\  before  0  has  regained  its  initial  value, 
at  which  point  -j-will  be  the  same  as  for  the  corresponding  point  g  of 

the  cycloid.  Above  the  point  g'  the  term  .  .  /  f{t) d.cosd be- 
comes negative  and  (with  its  negative  sign)  becomes  additive  and  there- 

dip 
fore,  above  g'  the  values  of  -7-  are  always  greater  than  for  corresponding 

points  of  the  cycloid.  Hence  the  angular  velocity  of  the  axis  can  never 
become  zero  and  consequently  the  axis  cannot  rise  to  its  initial  elevation 
and  form  a  cusp,  but  must  make  an  inflexion  and  culminate  at  a,  below 
the  initial  elevation. 

Commencing  a  second  descent  from  a'  with  an  initial  velocity^  the  suc- 
ceeding wave  will  be  flattened  (as  shown  in  treating  the  subject  of  "  initial 
gyratory  velocities"),  the  second  culmination  a^  will  not  (as  a  similar 
train  of  reasoning  to  that  just  gone  through  for  the  first  undulation  proves) 
be  as  high  as  ttj :  and  jpari  ratione^  each  succeeding  wave  will  be  more 
flattened  and  extended  than  the  preceding,  until  they  soon  virtually  dis- 
appear, and  the  curve  becomes  a  descending  helix. 

After  these  undulations  have  disappeared,  as  the  descent  is  only  due 
to  loss  of  rotary  velocity  (and  consequently  loss  of  deflecting  force) 
measured  by/(^),  it  is  evident  that  the  future  character  of  the  hehx  will 
be  determined  by  this  function. 

In  fact,  as  the  descending  velocity  —  is  then  very  minute  compared 

dip  ♦ 

with  the  horizontal  velocity  — ,  its  square  may  be  neglected  in  the  2nd 

equat,  (6);  and,  equating  the  values  of  sin  (9 -^  deduced  from  these 
two  equations,  we  shall  have 


By  differentiating  both  members  and  m^ng  various  reductions  we  get 


\Mgr 
S   A 


Mgy    __3sin2^~2_  (7 
A    Vsin^sin2^-2^^""-^'^^^^ 


an  equation  which,  after  the  disappearance  of  the  undulations^  gives  the 
value  of  d  in  terms  of  t. 

A.sf{t)  increases  d  diminishes  in  the  first  member,  to  the  limit  corre- 
sponding to  sin^  ^z=f  which  makes  the  numerator  of  the  fraction  in  the 
first  member  0,  and  the  denominator  a  maximum ;  showing,  to  that  limit, 
a  constant  descent  of  the  axis,  or  a  descending  helix  for  the  curve. 

As  the  values  of /(^)  hejond  f  (t)  =: n  do  not  belong  to  the  question, 
there  can  be  no  farther  descent  below  that  value  of  d  which  reduces  the 
first  member  to  zero ;  or  beyond  sin^^^  f. 


304  J.  G.  BARNARD  ON  THE  GYROSCOPE. 

At  this  elevation,  as  the  dejlecting  force  has  vanished  entirely  with  the 
rotary  velocity,  it  is  evident  the  elevation  of  the  axis  must  be  maintained 
by  the  centrifugal  force  alone,  due  to  the  gyratory  velocity. 

In  fact,  if  we  calculate  directly  the  angle  to  which  the  axis  must  fall 
from  a  horizontal  position,  in  order  that  the  velocity  generated  shall  be 
just  sufficient,  if  deflected  into  horizontal  gyration,  to  exert  a  centrifugal 
force  adequate  to  maintain  it,  we  shall  find  this  same  value,  sin^  0  =f  .* 

In  reality,  the  air  resists  gyration  as  well  as  rotation,  and  hence  the 
descent  will  continue ;  but  if  a  gyroscope  could  be  placed  in  a  perfect 
vacuum,  and  the  slight  friction  at  the  point  of  support  be  entirely  an- 
nulled, the  axis  would  descend  in  a  helix  until  it  reached  this  limit,  at 
which  it  would  forever  gyrate,  though  the  rotation  of  the  disk  would  soon 
by  friction  of  the  axle,  entirely  cease. 

*  If  the  solid  of  revolution  is  of  dimensions  so  small  that  it  may  be  considered 
concentrated  in  its  centre  of  gravity,  it  would  require,  in  the  fall  of  its  axis  through 

angle  90°-  9,  the  velocity  ^2  gycos^;  and  this  velocity,  deflected  into  horizontal  gy- 
ration in  a  circle  whose  radius  is  7  sin  6,  would  create  a  centrifugal  force  2  a  — — , 

^  sm  6 

008^9  , 

whose  component  normal  to  the  axis  of  figure  is  2  ^r  — — .     Equatmg  to  this  the 

opposing  component  of  gravity  g sin  9,  we  get  sin29  =f,  as  in  the  text. 

For  finite  dimensions  of  the  solid,  the  direct  determination  of  the  limit  in  question, 
is  more  complicated,  and  it  is  scarcely  necessary  to  introduce  it  here. 


ms 


..& 


